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.
Learn English – “Wait” or “waiting”
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"Little" also means "young."
From Oxforddictionaries:
(Of a person) young or younger:
my little brother
Similarly, you can call your older siblings "My big/older brother/sister."
Both, "young" and "little," are correct and can be used interchangeably, but I will be more inclined to choose "younger." Though "little" is more commonly used.
Do not say "small brother/sister," unless you are referring to their physical attribute.
Wait on is a colloquial variant of wait for ; it is not acceptable in formal discourse.
In my experience the two are practically equivalent, except that there may be some slight tendency to prefer on when the wait is caused by a delay.
ADDED:
Just to clarify: in Bieber's song wait on is a colloquial variant of wait for something or someone. There is also a formal use of wait on, now virtually obsolete except in historical contexts, meaning "attend, hold oneself in readiness to serve" a superior. This is (for instance) the sense you encounter when Samuel Pepys writes "Thence to White Hall, and we waited on the Duke", or in the title "Lady-in-Waiting"; and this sense is the source of the still-current use with servers in a restaurant.
Best Answer
Your examples express equivalent thoughts and both are used, I believe, with equal frequency.
Waitas a noun has the same meaning aswaiting, gerund.Of course, not all words as noun and gerund are exactly equivalent, just to confuse things.
Might mean a string of wins is broken
Might mean a race is finished
Could be a noisy clock or uncomfortable pillow cover
Could be that same loud clock or a blood-sucking arachnid
Sorry about that digression.