I sometimes use AUTO just to see what LR3 does to the photo. Then I adjust the photo the way I like it. You don't need us to tell you if you like the photo or not. Why are all these threads trickling into the "Beyond the Basics" section? This should be in the beginners section.
I thought this was Beyond the Basics? Is the camera algorithm or lightroom algorithm better?
Alright. There's the question, and it's a legit, good question. In part it comes from your observation of a discrepancy between the two algorithms. Start there. If the two software programs are returning different results then there are three possibilities.
1. The camera is right and LR is wrong.
2. LR is right and the camera is wrong.
3. They're both wrong.
I assure you the answer is #3. Neither algorithm is better and both algorithms can't do the job. Auto white balance doesn't work. Software can't see.
Earlier in the thread you expressed frustration about white balance. The first step toward beating that is to shift your expectations. You're looking for an "auto" solution in the technology and there is none. Frustration in this case is understandable. You've been misled to believe there's possible value in those auto algorithms when in fact there is none. In fact using them typically damages your photo.
Software tools can help, but the job is yours to do and, unlike software, you can see.
Best solution: Shoot RAW files and white balance manually. Using software to set a WB point in a photo can work very well. Good software will do a good job if you can hand it a known reference. I've been walking around my neighborhood lately taking photos. In nearly all my street scenes in the residential area someone has a TV dish on their house -- it's as good as a grey card. When there's no WB reference in the photo, the bottom line is: It's up too you. With practice you'll continue to improve. Software can't see -- you can.
Joe