is there any difference between "snoop around" and "stick one's nose in other people's affairs" as opposite to "mind your own business"? Is it possible to say politely "I would like not to snoop around" to indicate that I'm not going to insist on my participation = make smb's life hard this way?
Firstly, let's look at these idioms:
(The idiom) 'snoop around' action is intentionally done in order to find something secret or something about someone's personal life. The Free Dictionary describes it.
On the other hand, mind your own business is used when you don't want someone to ask or interfere about or in your personal life/affair. EnglishClub describes this with examples.
Stick one's nose in other people's affair means to interject one's self into another person's affairs. Implies that the interjection is unwanted -- OnlineSlangDictionary. As a side note, I have come more across poke someone's nose into.... as an alternative to this idiom.
Now your question:
The difference is snoop around is often carried out secretly generally to spy on someone or know the truth. If you snoop on a person, probably you are a detective and trying to find out the truths in that someone's life. A snooper is someone who makes uninvited inquiries into the private affairs of others. On the other hand, if you stick your nose in my life, you are trying to interfere in my life/personal affair. You are unnecessarily involving yourself in my matter. This could be someone's nature but the former one is not.
I'll try with examples:
I'm having doubts. I'll pay you for this. Just snoop around on my wife and find out who she's meeting every day.
Sure, you're good friends my wife. But let her make decisions on her own. Why do you stick your nose into her professional life? She's more capable enough to make a decision without your help.