"At vide" refers almost exclusively to something you "know" - something you've learned or been told. Formal or informal knowledge of something. "At kende" refers mostly to recognition... If you recognize a person, a place, a situation, a feeling, or a concept. This means that there is one bit of overlap: You can say "jeg kender til..." about a concept or something you've learned or heard, but usually this makes the most sense if you're telling someone that you know (e.i. recognize) something they are trying to tell you. So you might say "jeg ved hvordan det føles", meaning "I know how it feels", or "den følelse kender jeg godt", meaning "I recognize that feeling." In general, using "at vide" sounds more formal, and thus less humble. Telling someone you know something (or that THEY know something) is inherently insistent, and it can come off as arrogant. So generally, the safe bet is to use "at vide" when you're willing to be challenged on the claim. Meaning, if it's something you know, or you're pretty sure someone else knows it.