Quote is similar to reply only that when you click on it, it doesn't immediately take you to the input box where you quickly reply to the comment/post. Rather, it immediately adds the quoted comment(or selected text) to a quick clipboard that could be used later on within that thread by clicking 'insert quotes'.
For a single post, it's easier to use 'reply' but when you want to reply multiple comments/posts in the same comment(yours), that is when the 'quote' function becomes powerful.
It gives you the ability to reply multiple messages/comments/posts simultaneously in the same comment/replt.
If you see in a thread where there are many replies to different messages in the same thread, all sent by one user under a single instance of a reply, the 'quote' function was used to achieve that.