There are some topics that I won't stop talking about once I "get started", for example things that interest me a lot, like my hobbies. I will warn people not to "get me started" on such topics. It is meant as a warning to be helpful. I am trying to rescue them from having to listen to more than they want to hear.

They do seem to have almost the same meaning, but I think there is a shade of difference. "Don’t get me started" means something like, "Don't bring up that subject because it will make me angry/upset." Sometimes people say it comically. "Don’t start with me," means something more like, "Don't talk like that/do that to me (because I'll make you regret it)." The latter is something parents might say to their children. It isn't a harsh threat. Neither of these is very harsh.