To deal with this, I had to get out one volume of the Shorter OED. (It’s only of two large volumes, while the OED itself, if you can buy it in actual print, consists of a dozen or more books.)
‘Kin’ is straightforward in its meaning: it refers to the people you’re related to genetically. And in fact the Germanic ‘kin’ is related to the Greek ‘genus’ That’s a bit surprising, and not obvious, but it is in SOED. So that’s that.
‘Kith’ is less precise. The word is related to lots of words that have to do with ‘knowing’. Modern German ‘kennen’, to know – in the sense of have an understanding of or acquaintance with – is related, as is the Scots verb ‘ken’, as in ‘D’ye ken John Peel?’ … or whatever.
Kith and kin are the people you know, and the people to whom you are related.
Here endeth the lesson.