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.
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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.