Other than the writing, which I feel has stepped up several notches under Steven Moffat (even the Silurian two-parter was fantastic when I re-watched it, though I do still feel it was weaker than any of the other stories), the biggest difference is that 11 really cares. Yes, he gets frustrated, anyone with his intellect would, try explaining how the world works to an infant without feeling you're wasting your time. His heart, though, is truer than 10's.
The focus of my point is Lady Christina de Souza, at the end of the adventure in Planet of the Dead 10 was definitely ready to take her exploring the stars (only turning her down because he was still trying to recover from the loss of Donna). Lady Christina was a jewel thief, one who was trying to avoid capture when she was on the bus which went through a hole in space (one between London and Dubai).
11 would have refused Lady Christina's request for a different reason. 11 does not hand out TARDIS-travel as an escape from a life of bad deeds, 11 does not want friends who have no problems with breaking laws, 11 travels with people who deserve a reward for a life well-lived.
A similar idea revolves around Parting of the Ways and Amy's Choice.
In PotW, 10 reconnects with many of his former companions, all are now militaristic (Sarah Jane was not but then she was not really 10's companion, just a companion of 3 and 4 whom 10 bumped into). Life with 10 was not a lesson about life, it was an adventure to be carried on for as long as possible.
In Amy's Choice, Amy is torn between life in the TARDIS and a happy life with Rory after her travels in the TARDIS. While this happy life is just a dream, it is a perfectly plausible future for the Ponds. When, Amy and Rory leave the TARDIS, they won't be trying to recreate the life of action they had during their travels, they'll be putting childish things away.
