As quickly as possible can be translate as
Is there any difference between the two versions? What other adjectives have the second version (i.e., _________stmöglich)?
For your second question:
I selected a list of
schnellstmöglich means "as fast as possible"
möglichst schnell means "fast, if possible"
therefore, as knut already said, "möglichst schnell" is not as urgent as "schnellstmöglich" and more polite in that context.
This kind of phrasing arises from the superlative form of an adjective.
are the three states an adjective can have.
In this case:
By swapping these to phrasings, as you did, the superlative state just gets passed/swapped from one word to the other
Another phrase which might be heard is: als bald möglich or möglichst bald (I think my father uses this from time to time :) )