These are sometimes used together in a sentence - "he lurched and reeled around the room".
"Lurch" is a short, sharp, heavy movement forward. You don't get very far. "The car lurched forward, then the motor died."
"Reel" has a turning movement - like a fishing reel, but also the Irish reel (a style of music) has a spinning/rotating feel about it. "Thoughts were reeling in my head."