As nouns, are there any specifics situations for the use of each one? Or do they have the same meaning in all cases?
The wait is over.
The waiting is over.
Your examples express equivalent thoughts and both are used, I believe, with equal frequency.
Wait as a noun has the same meaning as
Of course, not all words as noun and gerund are exactly equivalent, just to confuse things.
The run is over.
Might mean a string of wins is broken
The running is over.
Might mean a race is finished
The ticking is annoying.
Could be a noisy clock or uncomfortable pillow cover
The tick is annoying.
Could be that same loud clock or a blood-sucking arachnid
Sorry about that digression.