There are a lot of good answers here, but I also wanted to add that "recover" can be an intransitive verb, whereas "regain" cannot. Used this way, it doesn't follow Harry's excellent summary.
In the emergency room. "How bad is it, Doc?"
"You'll recover."
"Apple made a blunder with their latest release."
"I'm not worried. They'll recover."
"Their marriage never recovered from her affair."