I think that it was a mistake, and perhaps somewhat arrogant and insulting, for David Tennant's Doctor to say that regeneration was like death, with a new man moving on.
David Tennant's first episode has him saying, "I am him. I am literally him. Same man, new face, well...new everything."
New everything means new personality and new way of looking at things. You are not the same man (or woman) you were at 18 as you are at 50. Your clothing choices are probably fundamentally different, and you might even like different food and drink (remember what you used to drink at 21? Still enjoy that crap at 42? Think 21 year old you would like the booze you enjoy at 42?)
But you are still the same person, just different.
Each Doctor has always insisted that regeneration is just a change of shape. Heck, most even insisted, despite evidence to the contrary, that the change was purely superficial. In fact, the 7th Doctor irritably snapped at Glitz "The change is purely superficial" when talking about his regeneration.
I think having 10 upset about regeneration is fine. But instead of the, "No! I'm the greatest! The next guy isn't me!" attitude they took, I'd have preferred something tied more into Donna, whom he called his best friend. Donna was someone he hated at first, but grew to be best friends with. Everyone else except Wilf seemed to HATE Donna too. Imagine the Doctor saying something like, "But even if I do regenerate my personality changes too. I like different clothes, different food....see things differently, you know? [looks at Donna through the window] What if I don't love then what I love now? What if what interests me now later bores me? I don't want that to happen."
That'd have been deep, and just fine. What I didn't like was the attitude that David Tennant was the best Doctor ever, and that RTD was the greatest writer, and that it will never be the same again. I felt that was showing in the End OF Time.
I also HATE that his last words were "I don't want to go." Puh-lease! Give me the 10th Doctor that I loved in the end.
You know what 10's last words should have been? "Allon'si! (let's go!)"
I'd have liked to see him, despite his fear, meet this new challenge head on, and with excitement.