They can be used almost interchangeably. "He should learn to leave well enough alone." "He should learn to let sleeping dogs lie." "Had he left well enough alone, the problem would not have escalated." "Had he let sleeping dogs lie, the problem would not have escalated." And yes, it means making something worse by not being content with they way they are now.

If you "leave well enough alone", or "let sleeping dogs lie" you don't try to improve or change something that's already good enough. Examples: 1.The kids seem happy enough now with their toys, so let's just leave well enough alone and forget about finding a new game for them. 2. Unless there's a problem, I'd suggest you just let well enough alone and let your staff get on with their work. "Leave well alone" , "let well alone" and "let it be" are also used, especially in Australian and British English. Yes, you can use the 2 phrases they mean the same thing.