you can only bother a person. It means disturb or annoy them in a small way by taking up time, asking them questions that they don't want to answer or don't have time to answer, etc.
Interfere means to enter into or take part in a matter that does not concern one, and in which one is not wanted: I never interfere between husband and wife/in his affairs.
If you interfere with sbdy./sthg. you get in the way of; prevent from working or happening: The sound of the radio upstairs interferes with my work. Hope this helps