They both mean pretty much the same thing, but I think "wrestle with" is a slightly more dramatic and less common way of saying it. They typically refer to long-standing and ongoing problems that somebody is trying to resolve. That could be, for example, learning German grammar (as in my case), or something more serious such as mental illness or drug addiction.

The meaning is very close, and it's hard to give examples of when you wouldn't really use them interchangeably. I'd say "struggle with" is less intense than "wrestle with". "Struggle with" is also more commonly used of the two forms. So: "I'm struggling with this homework essay" means "I'm finding it difficult or problematic." I don't think you'd say "I'm wrestling with this essay". You can "wrestle with your conscience" - when trying to make up your mind about, say, leaving a partner or job. Or maybe a friend has done something illegal and you can't decide whether to tell the police. I'd say that's a more suitable phrase than "struggling with your conscience", although that would also be OK. I actually think your characterisation is a good one: struggling is something over an extended period of time (I struggle with grammar; I struggle in social situations; I struggle with my alcohol consumption) whereas wrestling is trying to get something decided or resolved (I'm wrestling with a difficult decision). HTH

Just for interest, you can also say "battle with", especially if you're in southern Africa :-)

fighting with means physically or emotionally fighting something struggling is more of the pain of the fight if that makes sense