Core is one of the two versions of pywikipedia. It used to be called rewrite, which belies its interest in becoming a more efficiently-coded version of pywikipedia. However, because it is being actively rewritten, not all parts — or scripts — have been rewritten yet. Consequently, it has fewer working scripts than compat.
Should I use core or compat? Edit
As of 2014, most pywikipedia users on Wikia use the compat version, and it may be more convenient to download compat simply to have a shared experience with your fellow bot users.
However, if you're a first-time pywikipedia user, and you're not being actively mentored by a compat pywikipedia user, you should give strong consideration to downloading and using core. It's better-written, it contains most of the functionality you need, and it's the future of pywikipedia development.