To start, there's a common notion that the Protoss army is stronger while the Terran army is more mobile, so it's Protoss's responsibility to sit back, perfect their composition, and track Terran down to force big fights. That frame of mind is counter-productive. First, it's often not true. If Terran has time to get 3/3 upgrades, build a bunch of ghosts, vikings, medivacs, and replace some SCVs with MULES for a super-army, he's going to crush you in a straight fight. Secondly, the "which army wins if we ram them into each other" shouldn't matter at all if you play well.
The four pillars of late game PvT are:
Map vision which tells you where his army is, what direction it's moving, how many bases he has, and gives you advance warning for drops.
Mobility which lets you attack where he isn't and defend where he attacks. Pylons, warpgates and warp prisms give Protoss a huge mobility advantage over Terran in late game.
Positioning which lets you stay alive when you're behind and win engagements when you're ahead. Nail storms as Terran moves through chokes, and you stay alive even when you're 80 food behind.
Space control which allows you to limit Terran's mobility through key points. You can lock down zones of the map to defend efficiently or to punish Terran's positioning.
To illustrate the importance of these points, I'm going to start with a brilliant play from one of Parting's replays. In the following image, Parting finds himself in a situation where he's behind in army size, but is able to effectively use a combination of map vision, positioning, mobility and space control to destroy a Terran expansion for free.
Here, you can see Parting is down about 30 supply in army size, and both he and the Terran are establishing their fourth bases. Parting sees that the Terran is pushing his larger army towards Parting's fledgling expansion. If Parting were to sit idly with his army at his fourth nexus, there's a good chance he'd lose the battle and the game immediately. Instead, Parting takes advantage of his excellent map vision to force a trade where he knows he'll come out ahead.
Upon seeing the Terran force begin pushing, Parting immediately swings his army of zealots and archons toward the left side of the map staging a counter-attack. At the same time, he advances the four templar forward to control the ramp on the right side of the map (highlighted by a yellow line). He counter-attacks on the left while using just 4 units to control the space on the right. As it happens, Terran doesn't have sufficient map vision to see this development, so he presses the attack.
As you can see, Parting's army is moving to counter-attack down the left side of the map while two of his templar have moved forward to drop the first two storms. Parting will use the templar two at a time, and he spreads them so they can't both be hit by a single EMP. This play makes it almost impossible for Terran to deny storms. By moving the templar foward, Terran will be surprised by their position. And by sending them spread out two at a time, Terran needs four quick snipes to deny the storms. Instead, he eats the full storms.
Then, the next two templar (which were hidden to the right) show up and Terran eats two more storms.
At this point, a third of Terran's army is dead, the rest is in red health, and Terran recognizes that he'll be cleaned up by Parting's defensive reinforcements if he continues pushing. And now instead of threatening Parting's fourth base, Terran's army is simply grossly out of position for defense. While Terran's forces hustle home, Parting kills the fourth base and forces a lift at the third base. At this point, Parting simply pulls back, having turned a small deficit into a huge advantage without a direct army engagement.
So we know from that example how powerful templar can be for controlling space, and that we can use map vision to leverage that space control to gain an advantage in the game. Parting's four templar play to control the ramp was a creative and technical space control mechanism. Here's a less sublte way to control space that even noobs like me can pull off consistently:
You might see that and think, "Oh, that's standard base defense," and you'd be right, but with 3 cannons and 2 templar, you can block off any choke you want. Using just four supply (four supply!) you can make it tremendously inefficient for Terran to move through that space. The cannons prevent cloaked ghosts or marauder hit squads from threatening your templar, and the templar punish a full army bull rush. This method is noob-proof in that it works even when you screw up and don't have map vision.
Getting back to map vision and mobility, there are many other ways to leverage these resources to gain advantages. One of the best and easiest ways to increase your mobility is to put a pylon on the left side of the map and a pylon on the right side of the map, as close as you can get them to Terran's bases without them being spotted too easily. A warp prism bumps that mobility edge up another notch. As soon as you have options to warp in on either side of the map, you can do cute little tricks like warping some zealots in on the right:
And then hitting him on the left when he sends his army over to deal with your decoy.
These tactics are simple but effective. Remember that Terran can only kite in one spot on the map at a time and chargelots rock unmicroed MM. If there are three fights happening at the same time, kiting is only happening at one of those fights, which means you're kicking ass in at least two of them.
And if you augment the "hitting where he's not" tactic with a little map vision (controlling your opponent's watch tower for example) you can drop in his main (or if you're using pylons instead of a prism, send a pack of zealots at his third) as soon as he pushes out. I dropped DT's here because I had the tech and I knew he didn't have detection in his mineral lines, but chargelots would have worked great too. As it turned out, this Terran had tunnel vision on controlling his main army, so either drop cargo would have been game-winning.
It's worth noting that in those examples, I chose to attack with small forces while controlling my side of the map with my main army. This is in contrast to the Parting example I showed above where he attacked with his main army while controlling his side of the map with a small force. The right choice will be specific to the situation, but to get everything out of your late-game PvT, you need to set yourself up to be able to make that choice.
In the example where I dropped DT's, I failed to give myself the option of attacking with my army while controlling space with a small force to punish a counter-attack. Instead of keeping my army back at my third, I should have positioned it forward giving me the option of attacking down the right side of the map. Along the left lane, I could have stationed a handful of templar, perhaps with a few cannons forcing heavy losses on Terran if he chose to attack down that path.
This set-up would protect the attack paths toward both my third and fourth bases, freeing me to be aggressive with my army on the right side of the map toward Terran's third and eventually his fourth. It would also tend to funnel his army toward the action on the right side of the map, which limits his ability to deal with my mobility on the left side of the map with my warp prism.
Try watching your own games or pro games and thinking about how map vision, mobility, positioning and controlling space work together to determine who wins the game. Commentators focus so much on unit counts and compositions, but those are such tiny pieces of the puzzle. What matters most is who has more information and who better uses that information to score small victories before that final fight.