The only difference you'd notice is, assuming we're talking same class of drivers, the 12" will have a little bit more total output due to more cone surface area. Otherwise, with any decent subwoofers, no way would you be able to hear a difference from just a size difference. But especially if you were comparing the same family of drivers and the only difference is cone size, you'd never be able to tell a sonic difference until you reached the limits of the 10" and even then the 12" would probably start to show distortion too. There is none of this 10" will hit faster because it is smaller, magnets today are too powerful for there to be much difference. Again, this assumes we're talking decent to high quality subwoofers.
So with that said, if you can fit a 12" sub, do it because that will give you the most bass.
And in my business, I design and build home theater subs based on 18" and 24" drivers. They hit just as fast and articulate but with so much more authority than any 12" or 15" on the market. Then again, it helps when your driver has a 1.5 lb cone being moved by a 75 lb magnet and 4,000 watts of continuous power. The amp tells the magnet which tells the cone what to do; go, stop, it just does it so of course sound quality is great.