They are all synonyms although bother has a less formal connotation, you'd probably be more likely to say it to a sibling or child. Interrupt is only used in certain situations like sleeping or talking, where as disturb can be used in a very general way. These all are very nit-picky things and for the most part these words can be used interchangeably.

Bother and disturb can be used the same way, both means to irritate someone or make someone feel uncomfortable. On the other hand, interrupt is used when it breaks a pattern or stops something from going smoothly (eg. Interrupt my singing).