"I don't want to bother you" is usually used as a refusal of an offer. For example: a crowded cafe, a person sitting alone at a table with papers, books, coffee, laptop spread all over the table. Another person is standing, looking for a place to sit. Person A: "Do you want to sit? I can move everything and make room for you." Person B: "No thanks. I can see you are working, and I don't want to bother you." Person A is making an offer, but Person B doesn't want to inconvenience Person A. "I don't mean to bother you" is used as a polite acknowledgement that a request will probably bother someone else. For example: same cafe situation. Person B: "I don't mean to bother you, but can you move some of your things so I can sit down?" Person A might refuse or become annoyed but still move his things so that Person B can sit down. "I don't mean to bother you" is usually followed by "but", and then a request that might inconvenience the other person. It's a subtle difference, but that's how they are typically used differently.

They are the same.

They would be used interchangeably, there's no practical difference in use

Apryl has given a good answer. "I don't mean to bother you, but..." is a sort of apology: you're saying that although you don't mean/intend to bother the other person, you are going to do something that might bother them. In this case, you can also say "I don't want to bother you, but..." or "I'm sorry to bother you, but..." All three phrases mean pretty much the same thing. But Apryl has a good point about the other meaning of "I don't WANT to bother you." You can use this phrase to explain why you are not going to do something (for example, when you refuse an offer, as Apryl says). "If you're busy right now, I can come back tomorrow. I don't want to bother you."

The same