In a plot-driven story, the primary obstacles and struggles in the story are things that exist outside of the main character's psyche, such as the army of the big bad guy, or a murder mystery that needs to be solved, or a rival to be defeated. Pros are that they are generally considered much easier to write for an amateur, and some might say easier to read, and its easier, even necessary to stack obstacles which might be unrelated ("add more action", essentially). Cons are that emotional impact is minimal, there is little catharsis, and reader investment is low, it relies so much on 'cool things happening' that its difficult to hold interest for longer than the story takes to read/watch.
In a character-driven story, the primary obstacles and struggles are internal and/or intangible, such as succumbing to a desire for revenge, learning to love again, or wrestling with decisions regarding morality or duty. Pros are that emotional impact and reader connection is very high, and you can leave the reader with something much more lasting to think about. Cons are that it can be more difficult to write well, and you definitely have to be knowledgeable in terms of people and how they think, since internal processes are the core of the story.
Notice I said primary obstacles. The vast majority of stories offer both external obstacles and internal obstacles, but what is going to determine whether the story is more plot-driven or character-driven is what the core themes of the story are and where the author places his/her focus. Probably having a more or less equal amount of both is the ideal. Each type of obstacle/problem to be solved also has its own climax and in the best stories the climaxes come either on top of one another or even as a result of each other, further connecting the story and heightening emotional impact.
Personally I find primarily plot-driven stories supremely boring and all alike, showing a lack of imagination and story-telling ability, (I'm looking at you, every Epic Fantasy and Science-Fiction story written in the last 20 years!) Psychological stories are much more interesting and I enjoy that emotional catharsis that character-driven works give. This forum definitely needs more character-driven stories. Adding action is easy, its not something that needs much practicing, adding interesting characters is hard and much more impressive.