Earlier quoted context omitted.
In a sense, they're equivalent. RNN's create models of probabilistic state transitions. In other words, they create Markov chain models. You might argue that a Markov chain doesn't have memory, but it's easy to incorporate "memory" by expanding the definition of a "state" to include recent history or any function of the past.
No, they're not. And if it was "that easy" to incorporate memory RNNs would not have replaced Markov Chains As an example try to build a character level Markov chain for text generation and see how that goes.
It does appear that you would benefit from such an excercise yourself. No really, just try it.
A well trained higher order hidden Markov model would function no less impressively than an RNN and with good reason.