멍하니 is status without thinking. If you 멍하니 바라보다, I know that you don't have any thought.

물끄러미 is just status, so I don't know What do you think or feel.

There is no difference between 멍하니 and 물끄러미 in the meaning.

I think 멍하니 has more vacant nuance than 물끄러미.

As for me, 멍하니 reminds me of half-open mouth and 물끄러미 reminds me of closed mouth.  :)