The possibility to go for Free Investiture is the most important thing for which I want to declare independence or usurp a kingdom or whatever, because AI for some reason has never Free Investiture.
All the reasons mentioned here are valid, but I think the most essential one is that the bishops love you almost always more than the Pope when you have +25 relations with them. And that means troops and tax, although I usually have no church tax at all. That means +100 relations with bishops, max number of troops all the time and the bishops have more money to upgrade their churches by themselves. It's like the game is in Easy mode. Plus use Prince-(arch)bishops: super tame vassals, they always vote in your favour and never expand beyond their de jure borders or form factions or anything like that.
I don't think there is one reason not to have Free Investiture. The Pope sometimes complains about it but I pay him some cash or "deny on theological grounds". I never get an invasion CB or anybody excommunicated even when I have Papal Investiture so it is not any real minus in my book.