Now that you can get an Alta/Snowbird ticket to ski both mountains, and there's easy access back and forth once you're on the mountains, it's not really a decision anymore. There's a free shuttle that takes you back and forth between the resorts so you can get around in the evening too. Just figure out what kind of lodging you want and book it. If you're having problems finding exactly what you need, let us know what you're looking for and perhaps we can give you some suggestions.
We've stayed at The Cliff Lodge (Snowbird), The Rustler (Alta) and Alta Lodge. I have to say that The Rustler is one of the best places I've stayed ANYWHERE in ski country--courteous staff, excellent service, understated elegance, just like being in your own private club but not snooty by any means. If you can afford it, stay there (and remember that everything is all-inclusive, including gourmet meals, so it's not really as expensive as it looks if you like eating gourmet meals and staying in topnotch lodging). The food was excellent at both The Rustler and Alta Lodge. A friend of mine stayed at Goldminer's Daughter last winter and really liked it (he said the outside is rather ugly but inside it's nice). He said the food's great there too--and he's pretty discriminating, so I trust him on this. All of Alta's lodges include meals in their packages. If you don't want to be tied down to eating at the same restaurant each night, you might want to stay at Snowbird or in one of the condos. Nice thing about the lodges at Alta is everyone hangs out after skiing and swaps stories, you meet friends to have dinner with, etc. The Cliff Lodge was very nice, but like being in a hotel--very different from being in one of the lodges at Alta.
Personally, I think part of going to Alta is the experience you have by staying at one of the lodges. It's like stepping back in time when skiing was glamorous and exciting and a way to meet people who have the same passions.