There's a few issues here. Firstly, MOODLE_26_STABLE is the name of the 2.6 'branch'. We are getting into version control land when we don't need to but, essentially, this is the name given to the progression of changes applied to 2.6. Every week the stable version is updated with fully evaluated and tested patches and this will be 2.6.n+. The latest weekly is almost always the most up to date and stable version of 2.6 and the one I would go for.
MOODLE_263 is a 'tag'. In a way it marks an arbitrary point on the progression that is the 2.6 development where we decided the call it 2.6.3.
Ideally, have a read of - http://docs.moodle.org/26/en/Git_for_Administrators. If you set this up, you can get the very latest version of the 2.6 branch by (almost) no more effort than typing 'git pull' every so often which is cool