The two words can mean to obey a law, but there are other differences in meaning.
If a group has differences, but they make a compromise, we say that they will all abide by the decision, meaning that they will accept it without objection. We would never say that the members obey the decision.
Also 'abide' can mean tolerate, put up with.
"I can't abide his lateness" = I can't tolerate his being late.