"Refrain" carries a stronger suggestion that the thing you won't be doing is something bad, unhealthy, wasteful.
I'll refrain from criticizing my children.
I'm going to refrain from eating junk food.
I'll try to refrain from wasting too much time.
"Abstain" is common in two usages:
1. A legislator abstains from voting if he is present but does not support either side of a question. Often at the United Nations, the United States abstains on Security Council resolutions which condemn Israel.
2, Abstain, especially in the noun form "abstinence" often refers to sex. Abstinence has been recommending as a means of avoiding pregnancy or AIDS.
But as far as I can tell, abstain can be used as a synonym of refrain, with little difference in meaning. I feel that "refrain" carries a stronger suggestion that what you're not going to do is a bad or harmful thing. But
a peek at the dictionary and google convinced me that there's no clear rule on how to differentiate between the two, in either defining them or using them. Refrain is certainly the more common word.
Can our Anglo scholars add to this?