This isn't an in-depth analysis of the two hosting providers, just my limited experiences. I recently switched my WordPress blog from godaddy.com to bluehost.com, and I've been very pleased with the results.
- Bluehost seems way less gimmicky
- There aren't ads plastered everywhere in Bluehost
- They aren't trying to constantly upsell (in fact, there's really not much to move up to - they just seem to have the one plan)
- They don't charge extra for private domain registration
- The Bluehost interface is clean and uncluttered
- Setting up WordPress is pretty easy with either site, but it seems easier with Bluehost
- Bluehost's cPanel is easier to use
I've never had major problems with either site, so I can't compare the quality of their customer service. The customer support at both sites have impressed me the few times I've interacted with them (Godaddy's telephone support is really nice).
This isn't intended to bash Godaddy - they didn't treat me poorly, and I would have never tried Bluehost had I not heard such positive recommendations. However, after trying them both, I'll be sticking with Bluehost.
EDIT: 7.22.2010 I just had to use Godaddy's site to renew a domain & hosting account. It was a really horrible experience. It involves clicking through pages of annoying "special" offers, several new windows requiring different logins, and shopping cart items with no clear way to remove them (private registration - required using a different login at a different site). So I should state my point more forcefully. Don't use Godaddy, use Bluehost, they are way better.