Fucking... god damnit.
The Book: Radagast the Brown is not mentioned in The Hobbit outside of a few passing statements. Gandalf, speaking to Beorn, mentions that his "good cousin Radagast" lives in the southern parts of Mirkwood. Beorn remembers Radagast and refers to him as a good fellow; high praise for a misanthropic isolationist. Radagast is mentioned again in the Fellowship of the Ring at the Council of Elrond; he was the one who sent Gandalf to meet with Saruman, and who sent the eagles to rescue Gandalf from his subsequent confinement at Isengard. He is characterized as a "master of shapes and changes of hue," a man who speaks the tongue of birds and prefers the company of animals. The character is otherwise not described, and is never seen.
What does it matter? The mere inclusion of Radagast is an enormous difference, and easily one of the biggest and most controversial changes Peter Jackson has made. Radagast's involvement is important to the plot, and to the safety of the dwarves. This means, of course, that there are issues caused (or solved) by his presence that weren't in the novel at all. I will discuss each scene with Radagast in its own post, but the character himself has a very large impact on the plot of The Hobbit.
Regarding the character himself, Radagast isn't actually that different from how he's described in the books - but that's only because no more than a few lines are ever dedicated to him. Peter Jackson added many qualities and attributes to Radagast, but that's only because there was so much room there in which to add these qualities.
My Opinion: This requires some exposition: the five wizards are more than mere humans. They are Istari, a special kind of Maiar, which in Middle Earth are pretty much angels. Literally, they are spirit agents of the gods. Sauron, for example, is a fallen Maiar. They are incredibly powerful, intelligent, noble creatures. And Radagast has bird shit in his hair.
He always looks like he just sharted. Given the rest of Peter Jackson's portrayal of the character, this may actually be canon now.
I had very, very high hopes for Radagast the Brown. A wise, stoic wizard who disdains the politics of men, and who periodically gets involved only out of loyalty to Gandalf. What he lacks in Gandalf's charisma or Saruman's efficient logic, he makes up for in shamanism and animalism. A druid. That is not the case with the movie version. Peter Jackson has taken "eccentric" and "absent-minded" and turned it into full-fledged comic relief. He's goofy, in a Saturday morning cartoon way. He's more suited to be the mascot of some kind of sugary cereal than he is to be a wizard. I don't want to say Peter Jackson ruined the character (because, honestly, there wasn't much there to ruin anyway), but he certainly ruined a wonderful opportunity.