Brian's explanation is mostly correct. Bettering comes from the verb "to better" - to do things to make one a better or more successful person, but it isn't ALWAYS used when only talking about yourself. He was born poor but has worked hard to better himself.

To get better: X because of Y. Examples: I study to get better at English. I want to get better grades. I need to get better at listening. I hope to get better results next week. Bettering is used when you are talking ONLY about yourself. Examples: I go to the gym because I'm bettering myself. I go to church because I'm bettering myself spiritually. When I study, I'm bettering myself. It's a vague difference, but hopefully the examples help.