As nouns, quotation and quote can have the same meaning—referring to any quoted material:
She read me a quote/quotation from Henry David Thoreau.
In other uses, they differ slightly; only quotation can refer to the process of quoting:
There are many rules for academic quotation.
Also, only quote can be used to describe an asking price—the price being asked for something:
The seller’s quote for the house was much too high.
Only quote can additionally be used as a verb—with either of its meanings:
Donna quoted Henry David Thoreau to me.
The salesperson quoted too high a price for the house.
Finally, please remember that “” are quotation marks (not quote marks).