Let me know, is very informal used between friends. Example: If you are playing tennis tomorrow night, let me know so I can join you. Please tell me, is used when you are requesting information that you need. Example: Please tell me my test score. I wouldn't use either of these in a business email, I would use. Please advise. Or Inform me of the outcome.

I believe "Let me know" is former than the other way.

I think "Please tell me" sounds more urgent. "Let me know" is like a polite reminder. When they find out about something, they will "let you know." "Please tell me" sounds like I need to find something out right now. It's a more specific request. Hope this helps.