If 10 had regenerated immediately after having his rant at Wilf, I'd have been fine with it. I liked the way he acted with anger - he was scared. He thought he'd got away with it and he hadn't. I also like his final line, which I know a lot of people hated. The Doctor has always come across like the same man, but different when he regenerated. This was the first time it has been acknowleged as a change for the man himself. The idea that the man himself died and was reborn as someone new made sense for the character.
But it was completely ruined by the self-indulgant, unnecessary and totally stupid jaunt acorss the cosmos to say goodbye to all his friends. People who are suddenly killed don't get the luxury to wander around for ages, looking up all their long-lost pals and say goodbye. Way to kick the "it's actually like I die" motif in the face. It was also overly schmaltzy and felt way too much like it was more about RTD leaving than David Tennant.
Eccelston's was far more dignified and not drawn out to the nth degree. Personally, I also got the impression he was scared and didn't want to regenerate, but was putting on a brave face for Rose. Ultimately, as he gave his final farewell to her, it seemed as if he did accept that it was time for him and went out accepting of it. But then that was just what I read in it - which is also why I think it was the better regeneration. It didn't drag on and force-feed you the whole situation. It was quick, dignified and left most of the emotion of the Doctor to the viewers persoinal opinion.