I have not used either of them till yet. I looked at their websites and found their brief descriptions:
Github Gist:
Instantly share code, notes, and snippets.
Pastebin:
Pastebin is a website where you can store any text online for easy sharing. The website is mainly used by programmers to store pieces of sources code or configuration information, but anyone is more than welcome to paste any type of text. The idea behind the site is to make it more convenient for people to share large amounts of text online.
Apart from the explicit mentioning of "anyone is welcome to paste any type of text" in Pastebin's description, I see no difference in what they are used for.
I'd appreciate if someone could point me the differences between them.
Gists are way more 'advanced', since Pastebin is very simple and user-friendly. At very least, gists are used to be shared mostly with coworkers, while pastebin is heavily being used for wide-usage or for private usage. https://help.github.com/articles/forking-and-cloning-gists/