not much, really. Skinny might seem to be more judgemental, a less flattering thing to say to a person than to say they are thin. This dictioanary site is good for looking at a range of definitions and usages of words - http://www.onelook.com/
"Skinny" is usually used in more extreme cases. If someone is bony, I call them "skinny." Any higher amount of flesh will make me consider them "thin." Generally, "skinny" is not a compliment to most adults. If you want to compliment someone, call them "thin."
Now, in colloquy, "skinny" can be used with great positivity, such as saying, "My goodness, look how skinny you are!" However,