Facebook...this is for social networking, being friends with people you want to be friends with. You can also upload pictures, comment on other people's status and pictures, etc.
Twitter...this is usually a quick one sentence update to let people know what you are doing. People can follow you without your permission and you can follow as many people as you like without their permission.
Depending on what your lifestyle is, you may prefer one over the other. If you want to be in touch with more just your friends, Facebook is your answer (since you can limit who can see what). Twitter is more of an open playground where a post can be seen by anyone and everyone around the world.
I personally prefer Facebook.
I have both a Facebook account and Twitter account, and this is my own personal preference.