It has been said before (by Blizzard) that Carriers and Tempests have a lot in common, and that they have yet to see an argument for keeping Carriers in the game. First of all, to some extent all air units (and even all units) have similarities. Banshees and Carrier have similarities, for instance, in that they are all air units which deal damage and therefore are good against units which can't shoot up. However, Banshees can cloak and deal high dps, while Carriers are more resilient, better in the endgame, and create interceptors. So what we're really talking about - when we say roles - is their comparisons to other units (in particular to those which they are meant to be effective against and those which they are meant to be ineffective against).
What I intend to show below is that in terms of reactions from the opposing player, in terms of viability early/late game, and in terms of uses on the field of battle, Carriers have a lot of things Tempests do not have. In some respects, the Carrier is protoss's Thor - and I don't think it'll be fully understood until the stargate buffs are through and we get to see what they can do when Protoss is more able to build them. Blizzard didn't remove the Thor, and I don't believe they should remove the Carrier either.
The response of the Carrier itself, as depicted by this cat, ought to be enough:
Just give the little guy another chance.
Why are we not building Carriers in WoL?
First let's talk about the Carrier. Undoubtedly, the Carrier (with other stargate units pretty high on the list) was the least used protoss unit in WoL. Why? This is no mystery.
First of all, stargate as a tech path requires detection from elsewhere - so you need to build both a stargate and a robo in PvP and PvT (and possibly now in PvZ - tho Swarm Host timings are still relatively new).
Second, the 120 second build time put the Carrier in a situation where it usually would not come in high numbers until very, very late in the game when the opponent already knows what you have and is given time to react to it.
Third, generally protoss armies are built around a core of gateway units - which generally have lower dps (for cost) than their equivalents from other races. Units from the Robotics Facility or Templar Archives, on the other hand, are longer range and have high dps and therefore synergize well with the gateway units.
In PvT, early and midgame bio pressure is quite heavy and the Carrier lacks adequate timing (due to a 120 second build time) to stop many 1-base pushes, and are fairly late to the party against 2-base midgame aggression (assuming your scouting info made you build a robo first). In addition to this, Carriers have difficulty in the midgame against both Vikings and Marines - both of which tend to be accessible in nearly all Terran openers - with a brief timing window where they shine if not scouted (most of the time T has had too much time to build viking numbers or marine numbers to eliminate Carriers or interceptors respectively). They can also be a reasonable lategame tech switch.
In PvZ, Carriers are becoming increasingly popular in the endgame due to their efficiency against BL/Infestor and since you need their costly tech path anyway for Mothership - but in the early/mid game, certain timings make Carriers seemingly impossible to wait for (not least of which is the 12 minute roach max).
That's why they weren't used - it was a combination of long build times, low dps for cost, and lack of detection for the tech tree.
How Carriers would be better served in HotS
In HotS, one of the big problems was taken out - protoss have access to detection through the Oracle (whose cost is currently higher than Robo+Obs, but that's the subject for another post). In addition, some of the more potent new early/mid game aggression we are seeing is best responded to through Stargate units (looking at Warhound and Swarm Host).
All of a sudden, upon scouting certain things, Stargate looks not only viable - but like the best choice, so we're closer to getting Carriers anyway in the early/midgame.
And building carriers forces things like Vikings and Corruptors, which may seem less readily available in HotS (due to medivacs not synergizing well with mech, and since Mutalisks have not gotten any stronger in HotS, and thus Spires may come later).
Comparison of the Carrier and the Tempest
As I said above, one of the Carrier's big weaknesses is its low dps. Colossi and High Templar are both very high dps units due to their AoE abilities and long range - while the Carrier (at just 96 damage every 3 seconds with +1 attack researched) is more in line with the Stalkers dps/gas expenditure ratio, while the Colossus and HT are often much higher due to their splash damage. The biggest thing I hate about the Tempest is that with +1 researched, it is less than 1/3 this dps - and costs more gas (9 dps for a 300 gas unit - seriously...). This means that, in order for the Tempest to have a higher dps than the Carrier in army-to-army combats, it must have approximately 3 times the opportunities to attack. This is unlikely, as (like the Carrier), it's also slow, so units like the Viking and Corruptor can just chase them down and attack them unless some higher dps units (or photon cannons) are going to prevent them from doing so.
The sniper rifle from Starcraft: Ghost might look scary, but in order to balance its range, they've made it so that you have to actually shoot someone a dozen times in the head to kill them with it. After all, it creates a much better game dynamic when you can get shot in the head three or four times and say "you know, if I don't do something about this in thirty seconds or a minute, that sniper's gonna kill me!"
In addition, due to this low dps, it makes players who get Tempests in the early/mid game more vulnerable to early pushes (since they're also less dps/build time - not just less dps/cost - that's how low their dps is).
Also, the Carrier, because it does much more dps, is much more likely to require a reaction from the opposing player. As I said, earlier, Carriers force things like Vikings and Corruptors. In comparison, a Tempest, for instance, will take about 50 seconds to kill their food cost in Marauders or Roaches (which cost much, much less, especially in gas) - and, therefore I expect in many cases the reactions to a bunch of Tempests in the early/midgame will be "do what I'm doing anyway and pretend the Tempests aren't there" - especially so when you consider that it takes a Tempest 80 seconds to kill its supply in Warhounds (three of which cost much more comparably to the Tempest). Sure, you can also get a Starport and a bunch of Vikings and try to snipe it - but with the Tempest's dps, you also really don't need to if you don't want to. And on top of this if you have 2 SCVs along (per attacking Tempest) you can merely turn on auto-repair and essentially ignore the damage done, as each SCV repairs about 5 health/second (unless they have 8 or more Tempests - that's 48 supply - focus firing a single Warhound - but my point is made, the damage is so low it's silly). Carriers require something like 6 SCVs per attacking Carrier before you can ignore the damage - and only 3 are needed to 1-shot a Warhound (or 2 if for Marauders/Roaches with the appropriate upgrades).
My point here - since we're talking about the carrier, not bashing on the Tempest (which again, may be the subject of another post) - is that Carriers can make the Stargate path more viable in the early/midgame - which is where Stargate has had issues historically, and therefore adds soemthing to the Stargate tech path. Carriers also require a reaction in the lategame from the opponent - Corruptors or Vikings, whereas all but large amounts of focus-firing Tempests can be ignored or repaired through, and large amounts of Tempests, no doubt, will run into problems as they are supposedly a support unit.
The carrier is a stronger, more resilient Tempest which is useful in direct confrontations - something the Tempest and Protoss air in general is lacking.
Comparison to the Void Ray
Many people may be asking - if you're talking about dps/build time, earlier game scenarios, and dps - why not just build Void Rays? They're higher dps per just-bout-any-metric.
This is true - Void Rays have a ton of dps compared the carrier. This isn't why people don't build Void Rays, though. People don't build Voids because they are supremely weak to lategame units like the infestor, the thor - which are easily able to pick of large squads of them with splash damage. Carriers don't have this problem. They're more resilient and (since they're not the unit which is attacking) have more relaxed positional requirements. In addition, they have an easier time with vikings (which voids have great difficulty catching up to, while it's much easier for Carriers - and once they shoot, their interceptors remain around) - albeit, they are still not great against Vikings, but they're not as awful. And their high armor make them difficult for Marines to target down.
Not Pictured - all the other Void Rays which died to let these few finally get into firing range.
Add to this the fact that their interceptors function as no small number of hitpoints for their opponents to get through, and well, its not hard to see that if you're able to get them out, they're just better in many situations than Voids are. Given an upgrade lead, they deal comparable damage, and are much tougher to kill.
Conclusion
Carriers are more than just a fun unit that people like for sentimental reasons. They're more than just a long range flying unit, and they're more than an end-of-the-tech-tree. Carriers are strong, powerful units which have the potential to be useful in many matchups given the newfound availability of detection (from the Oracle in HotS). From Day9's Carrier Funday (and the resulting HuK submissions, which made just about everyone try Carriers early game for a short time at least) to HongUn's Carrier build in the GSL against DRG to MC's recent Carrier PvT timing push, the unit has seen various experimental uses in WoL even despite the unpopularity of Stargates in general. And some of them have worked out for those who have tried them - even recently and even at the pro level.
Wait a sec, MC teched to... what?
As we've said, once Stargate has detection, fast carriers (ok, not 1-base fast, but reasonably fast) has the ability to be a build that is seriously viable. We've already seen timing pushes working in the pro scene in the midgame (MC vs Kas in IEM). Maybe this will be the same with Tempests - but I have a hard time believing that due to its low damage. Lategame carriers have something Tempests don't have as well - the ability to slow push and flatten things like Mech (which are becoming more prominent in HotS), roach pushes, and other things that can't shoot up which quite frankly can live long enough to ignore the Tempests' paltry dps. The Carrier has a serious place in direct engagements, both as a source of dps which is hard to kill off (due to resilience, not range) and also as a source of hitpoints to draw out battles and give things like storm and Void Rays (and Remotely participating Tempests?) a real chance to get damage going (little known fact - interceptors are the best conversion of minerals to health in the game - provided you don't lose the carrier in the process of deploying the interceptors).
Few who have tried seriously massing Carriers have come away from the experience thinking "man, Carriers are bad against everything once I have them" - if anything, they've come to the conclusion that Carriers are a unit that's just too hard to get. Blizzard is making it easier now to get them, right as they're removing them. It doesn't have to be this way. The Carrier wants a shot and it feels like it's had enough fringe uses that beefing up the Stargate tech path should be enough to push it mainstream. I sincerely hope Blizzard doesn't let endgame Stargate play hinge on something new, unknown and possibly just as fringe as the Carrier was in HotS. Instead, let the Carrier be the big beefy endgame tech option we always wanted it to be.
Alternatively... An Addendum
I'm not sentimentally attached to the Carrier. I don't have visions of interceptors dancing in my head. I don't think back to BW and say to myself "it's no good if it's not like that!" If it's going to be the Tempest or bust, change the Tempest. All of the things I'm describing are just as applicable to the Tempest if it's made into a combat unit instead of a "long range sniping unit". I like the idea of the Tempest, but it just doesn't fit with Protoss, and I don't mean that stylistically. I mean that the early game is too vulnerable for Protoss to open up and try to make a bunch of units which don't deal damage very well, and are unable to respond to midgame aggression. We do need a long-range unit to deal with things like BL/Infestor, Tank/Marine, and other such things - but it certainly doesn't have to be 22 range, and it doesn't have to be so sturdy. Protoss gets units like the Immortal and the Colossus and the High Templar so early not because they love the style - but because they need those units if they want to withstand aggression - and they need those units if they want to get aggressive, too. If it must be the Tempest, then make the Tempest strong like the Carrier was (once you got one), give it the potential for high dps, the potential to stand up to things like Thor splash and Fungal Growth, the potential to hit Vikings and Corruptors (I don't need them to be cost effective, but when you see things like VR vs. Viking - you know somethings off), the potential to take marine fire - the potential to wreck other artillery like Siege tanks and Broolords. Make them a unit to be feared the way I get afraid when i see an army of Broods, Siege Tanks, Battlecruisers and Infestors. I'm not seeing why you'd need to fear them right now, because (at least in the beta) I haven't seen them doing anything to turn the tide of battle the way a tech tree-ending unit like the Carrier could - at least, once you got them.
If you like the Carrier, or want to see the Tempest get some love, I have a similar post up on the bnet forums at: http://us.battle.net/sc2/en/forum/topic/6521203710