They are very different. 'Peer' is a formal word, often used in scientific and academic contexts. A peer is person of the same age or status as you. A 'mate' is an informal word meaning friend, used mainly in British and Australian English. It's a rough equivalent of the American word 'buddy'.

"classmate" carries more information than "peer". Classmate is a friend from your CLASS but a peer can be your friend from your school (not necessarily from your class) or just someone who is your age. "mate" is a synonym for "buddy", "bud", "friend", "pal" etc.

Hi Mimi! Peer is a person of the same age, the same social position, or having the same abilities as other people in a group. For example: Getting help from a peer is easier than asking a teacher. Mate is a friend or used as a friendly way of talking to someone, especially a man. For example: We've been mates since our school days/Have you finished having your dinner, mate? Hoang Kha

@Gavin. I liked it when I was there. It sounded friendly. :)

In Australia, 'mate' is quite common, and isn't necessarily a male thing, it can be for females too. Like SuKi says, it's similar to "buddy". Note that it originally meant friend, but sometimes people use it for other people who they are not yet friends with. These days it can be used for anyone.