They do mean similar things but I would say 'bug' is more of a slang term which hints that you are annoying a person - so like don't annoy me, and bother means roughly the same but is perhaps less informal. Disturb is like interrupt but they can not always be used to replace the other. If you cut into someone's conversation, the person may say "Don't interrupt me" but they could not say do not disturb me. Do not disturb me is mostly a request to be left alone in peace and quiet or to leave them alone as they are busy doing some other task. "Do not disturb me while I am in my office writing." To use 'interrupt' instead would sound odd to native speakers, to use the other words would suggest you found the person annoying, "don't bother me' is somewhat more rude and informal. Hope that helped!