I guess they're pretty similar. However, if I think of a Markov regime switching model, I usually suppose that the observable state y is a function of both the unobservable Markov process and lagged values of y. That is, I think of a time series process in y such that the coefficients are perturbed by a Markov chain.
On the other hand, most HMMs I've seen have y as a function of the unobservable Markov state only.
However, usage changes across time and fields. Other readers might differ.