It's hard to generalize too much- Hsu & SVS make some 10" subs that will stomp all over some other companies 12 inchers. Obviously they don't rewrite the rules of physics, but they use drivers with large & powerful motor structures in well designed ported boxes that milk every ounce of bass to be had.
I'd say a 12" won't necessarily play any
deeper/lower than a 10", but all else being equal it will play
louder. A good example is the old 8" Cambridge sub I had- it would play about as deep as the larger 12" version, but it would bottom noisily if driven to high volumes. At low volumes they were subjectively about the same.
Whether or not a 10" driver will satisfy you depends on a lot of things, like how loud you play it and how much bass you like. If you're a "basshead," you may as well go with a pair of 12 inchers. If budget is a concern, I'd certainly advise that a good 10 is better than a cruddy 12.
However, you could get a 12" Adire Rava for about the same price as a lot of companies 10 inchers ($400 + shipping), so if you have the room you might be better off going that route.
I'd suggest you do a little reading on the Adire, Hsu & SVS websites. There are plenty of other companies out there that make good subs, but those three are legendary for their value & performance.