Uptown's where it's at! Dinkytown is not so much where it's at.
Live in Dinkytown if you: are under 23; were a member of a Greek organization in Undergrad; plan to never cook and always eat out (i.e. no grocery stores); Don't mind being surrounded by 90% Undergrads and all that that entails; Want to be within walking distance of college bars & restaurants.
Live in Uptown if you: are 23 and over, or mature for your age; Want to be surrounded by a diverse type group of people i.e. Young yuppies, Young hippies, Young blue-collar workers, Young artists and musicians, Grad students, non-white people, etc.; Want to be within walking distance of an enormous variety of bars & restaurants and shopping.
Pros of Dinkytown: Walking distance to class, or a 5 minute ride on the Campus Connector.
Cons of Dinkytown: Overpriced. Surrounded by Undergrads (some will consider that a Pro)
Pros of Uptown: Better value for your money, 10 minute express bus ride nearly directly to the law school's door (I think grad students can get reduced bus passes as well). Wide diversity amongst night-life options and neighbors.
Cons of Uptown: Maybe some people are averse to taking the bus? Other than that, there's really no cons to Uptown.