ved (vide): have sikker information om noget; have erfaret at noget er sikkert eller sandt (to have certain information about something; to have experienced that something is certain or true).
kender (kende, and this is just one of the definitions): vide hvem nogen er; være personlig bekendt med (know who somebody is; be personally familiar with).
It's funny how "vide" appears in the definition of "kende". But generally "vide" is about something brain related, where "kende" is related to pretty much everything else.
Some guy also attempted to express the difference, found here: http://jesperhansen.co.uk/vide-or-kende/
But I realize that this is not trivial, and even though they may seem interchangeable they aren't.
I remember reading some where that the English language also used to have two words for to know. 1: to know and 2: to cun Cunnen is still used but not as a verb anymore but it still has his adverb cunning. Other languages like my mother tongue Dutch have also to verbs for to know: weten and kennen. Just like french: savoir and connaitre. It is a common thing, with English being an exception
For interest, in some areas of Scotland we use the word "ken" to mean "know", e.g. "Aye I ken" meaning "Yes I know".
I believe "ken" is derived from the Old English "cunnan" like you said, Rogerwtje, whilst "know" is derived from "cnāwan". So the alternative word is not completely lost if you turn to Scots!
I'm going to take the liberty of translating from this website. For the verb "kende" (know) we have the noun "kendskab" (familiarity). For the verb "ved" we have the noun "viden" (knowledge). Normally "viden" is perceived as a deeper insight than "kendskab", which can perhaps be more superficial. In practice "ved" often takes an object that is a complete sentence, while "kende" often takes an object, which is a pronoun or a noun.
"Jeg kender hans adresse" (I know his address) /// "Jeg ved hvor han bor" (I know where he lives)
"Jeg kender ham godt" (I know who he is) /// "Jeg ved hvordan han er" (I know how he is - in this case we're not talking about well-being, but what kind of person he is)
"Kender du dansk kultur?" (Do you know Danish culture?) /// "Ved du noget om dansk kultur?" (Do you know anything about Danish culture?)
Hope that helps!