It's unusual to say "fix me" or "mend me", unless you're trying to use metaphoric wording. That said, here are some nuances:

1. We fix things that are broken. So if you said "fix me", I would take that to mean that you are damaged, such as psychologically or physically. I can see how "fix" can be used for psyciatrist treatment or for a bone fracture (where you need a plaster cast or some other "setting" device to help the bones grow back in place).

2. We mend things that are  injured or damaged, but not irreperably. For instance, when you repair/sew a hole in a sock, you are "mending" the sock. When you are recovering from something (like an illness), there is an expression "on the mend" to mean that you are recovering from that illness.

3. We correct mistakes (actions and/or things that are wrong). For example, a child learning to write confuses 'b' and 'd' and writes 'deb'. You correct him and write "bed".

By the way, the question should be "What is the difference?"