What's the difference between Reply and Quick Reply:
Quick Reply is made as a "Quick Response" block, so you can make a response to a post (or thread) before you forget or as you read them. Quick Reply is meant as a "no thrills" response box, as it's just there for a fast/non-formatted response.

Reply however brings you to a "Full Response" page that allows for heavy posting.

7.2 will introduce a feature which will allow you to click a "Full Reply" button from the Quick Reply box which will forward you (and all of your Quick Reply text) to the Full Reply page.

What's the difference between Quote and Quick Quote:
Quote will quote the users text to the Full Reply page. Quick Quote will quote the users text to the Quick Reply box below the thread.