'Abstain [from something]' is quite old-fashioned, and often has a moral connotation. 'Abstinence' makes you think of monks abstaining from sex, alcohol and so on. It's usually used for a period of depriving yourself of something which is enjoyable but considered unhealthy - hence the moral overtones.
Aside from these associations, the main use of 'abstain' nowadays is when someone declines to take part in a vote. If you abstain from voting, your vote counts as an 'abstention'.
By contrast, if you refrain from doing something, you momentarily curb your inclination to do something. You might disagree with someone's opinion, but refrain from commenting. This means that you wanted to give your opinion, but you kept your mouth shut for the sake of being polite. You couldn't use 'abstain' here.
'Refrain [from something]' tends to be used in short-term contexts. For example, you might see a polite sign asking people to 'refrain from smoking'. This means that they are being asked not to smoke when they're in a particular location out of consideration for others. This is different from 'abstain from smoking' which sounds longer term and has a moral and/or medical association.
If I were you, I'd avoid using 'abstain', other than in the context of voting.