Hi Summer - I would say these are essentially the same thing.  However, "on the contrary" is probably 10x more commonly used than "conversely."

I'm not super grammatically knowledgeable but I can tell you that conversely to me sounds like you're going to talk about something opposite. While saying on the contrary kind of has an argumentative tone to it, as if you're going to give a rebuttal or something.

In the end, there's such a small difference that even us native speakers who don't study our own language's grammar don't know, so don't worry about it too much.

But I do agree with Patrick, at least for normal everyday conversations you'll likely never hear someone say "conversely".

Thanks a lot, Patrick and Matt.