Is there a difference between "quicker" and "more quickly"

Solution 1:

Quicker is the comparative of quick, which is an adjective; more quickly is the comparative of quickly, which is an adverb.

Informally, quick is also used as adverb, with the meaning of "at a fast rate, quickly." These are the examples reported by the NOAD (third edition).

He'll find some place where he can make money quicker.
Get out, quick!

Solution 2:

I think quicker is sometimes used as an adjective, but as an adverb both are used to the same effect. From Google's Ngram, it seems that quicker used to be more common but that more quickly is slightly commoner now.