Goofy is a dog who wears clothes, drives a car, and talks. Pluto is a dog that basically acts like a dog. Why the difference?
In general, the obvious: Goofy is anthropomorphic, Pluto is not. In effect, for plot purposes, you can think of Goofy as a human, and Pluto as a dog.
It goes deeper than than, but you have to look at their origins a bit.
Pluto was based upon the Bloodhounds from the movie 'Chain Gang', the only significant difference being the color of his ears. Like his inspiration, he's a dog, first and foremost, and his role was to be a dog. As such, he was able to act as a foil to other characters, serve to move the plot as needed due to doing 'doggy' things, and so forth. He was a minor character, until 'Playful Pluto', where he finally took a starring role.
Pluto also didn't speak. He could bark, growl and so on, but he couldn't actually verbalize. As a result, the animators were forced to portray his thoughts, moods, etc., via body language and facial expressions. (At least once, they cheated somewhat, by having an Angel/Devil pair on his shoulders that could speak, but this was unusual.)
Pluto was clever (sometimes), but he was still just a dog; the situations he got into were those that his level of intellect was appropriate for, and he had only the tools that a dog had for resolving them.
Goofy was an anthropomorphic dog. He could speak, he had some degree of reasoning and intelligence, and, in effect, the 'dog' aspect was never significant. Much like Donald Duck, Mickey Mouse and others, he was, in effect, a human with various animal inspired qualities to his appearance. In Goofy's first cartoon, he was referred to as 'Dippy Dawg', and was shown as basically an annoying rube annoying others at a performance.
As with Disney's other anthropomorphized animals, he got into situations that you would normally see a human get into. He spoke to others, he used his intelligence (or, often, lack of it) to move the plot along, and was able to do anything a normal human could.
If you look at the plots of situations they were both involved in, they really aren't very similar; Goofy is in plots where a human could readily have been substituted, and Pluto is in distinctly animal plotted cartoons.
From the Wikipedia article on Goofy:
Disney has needed to deal with a certain amount of confusion concerning the fact that the anthropomorphic Goofy is treated as a human while Pluto (an ordinary dog) is treated as a household pet, despite being of the same species. On their web site, it stated that "Goofy was originally created as Dippy Dawg" and "was created as a human character, as opposed to Pluto, who was a pet, so [Goofy] walked upright and had a speaking voice".
Why did Disney chose to make Anthropomorphic animals as their stars? It could be for any number of reasons, including avoidance of offending any specific human group (unlikely, given the era, but possible), making them more attractive to kids, and so on.. But that's another question. :)