Great question! Bother/disturb: Rather than formal/informal, the context of when to use these words is rooted in severity of a situation. For example, 'bother' is typically used in mild contexts (i.e. when little siblings bother/irritate older siblings causing a nuisance). 'Disturb,' on the other hand, is a few degrees higher than 'bother.' If a situation elevates from bother to disturb, it literally means that one person's ill actions have completely affected another person negatively. Hopefully you comprehend the differences, now we'll compare the similarities. In the sense,"Don't bother me now" / "Don't disturb me now," the words have very similar meanings, but both words can be used in other ways. "It's no BOTHER" - "it doesn't BOTHER me; I don't mind" "Don't BOTHER [doing something]" - it's not important enough to worry about, or to BOTHER WITH DISTURB generally means to disrupt something that was previously calm or settled. DISTURBING describes something that affects you negatively emotionally, for example "The live execution of the prisoner was DISTURBING". Another gradation of this word set would be: irritate, agitate, or annoy. Best wishes!

In the USA, hotels usually have "Do Not Disturb" signs that guests can put on the outsides of their doors to keep the maids and others out. The signs never say "Do Not Bother".