The full find takes you to a tabbed layout where you can find contacts by just about any criteria. The Quick find leaves you on the Contacts page.

When you press the Quickfind button you will see thevarious fields go blank. You can then use any of the fields, name, email, phone etc to find the record or records that match.

Note that you cannot find based on payments or items from the Quick find. For that you must use the full find.