I guess it goes without saying that The Vatican is going to house the greatest and most impressive collection of ancient Roman and high renaissance art in the world. After all, these are the guys who commissioned all the work. But I'll say it anyway: The Vatican houses the greatest and most impressive collections of ancient Roman and high Renaissance art in the world!
The Sistine Chapel alone is worth a visit to Rome! Unfortunately no photography is allowed when you're in the actual chapel but don't worry - I'll post photos.
Add to that ancient sculptures like the Laocoon, the Belevedere Torso and the Apollo Belvedere... and then Raphael's frescos (like the School of Athens); and Michelangelo's Pieta in Saint Peter's Cathedral (which itself is a wonder of the world) and you've got perhaps the epicenter of Western art, culture and artistic expression.
It's also the busiest place I've ever been. I've visited a few times on separate occasions and there's no "good time" to go. No "off season." It's mobbed every day.
However it's possible to go on private, guided tours both before the museum opens (you meet at 6am and enter at 7am) and also after it closes. There are also private tours of St Peter's Basilica (you climb up a hidden staircase inside the basilica and up into the cupola where you can see down into the church, or step outside for the best views in Rome). They're a bit expensive but - if you possess a certain level of admiration - well worth the extra few hundred dollars, over the course of a day, to have private access to things like the Sistine Chapel with just a handful of fellow tourists. Having visited amongst the throngs twice before, the 3rd time I sprung for the private tour (we went at 6am) and I don't regret it at all.