Both are great games, but it really depends on the platform you intend to play on. If you're thinking xbox, then I'd go with Fallout, and for PC I'd push Bioshock. Fallout3 is going to be the bigger game just because it's more of a sandbox shooter, but Bioshock has a great story and some replay value (due to its multiple endings).
It also depends on if you like the Bethseda style RPG. I've bought several of Bethseda's games (Oblivion, Morrowind, and Fallout 3), and I always get lost in the game because they are so massive. If you like that style though, it's a good bet you'll like Fallout.
If you prefer linear games that have objectives and frantic gunplay, Bioshock is really hard to beat for the money. But again, if you don't like scary moments in your games at all, then Bioshock probably isn't the best option (it's not terribly scary, but there are some eerie moments and the occasional enemy out of nowhere jumps).
But the best option (applies to PCs mostly) is to find good deals online. Steam has amazing deals for PC games at times, and other sites like direct2drive.com and gogamer.com have some pretty nice deals for sales as well.
And btw, Fallout3 is 25 bucks on Steam right now, so that would be a decent investment.