Both the words Refrain and Abstain basically mean to avoid or to stop oneself and hold back. However there is a little difference in their usage.

One refrains from an action;
We all must refrain from oppressing the weak and the poor.
( oppressing is the action we should refrain from )

One abstains from a thing;
All sports players are directed to abstain from any kind of drug use.
( drug is the thing players should abstain from )