I'm going to take the liberty of translating from this. 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!