"Kennen" is an (intimate) knowledge of something whereas "wissen" refers to your fact-keeping capabilities ;) That's why "kennen" is often used for people, emotions, experiences whereas "wissen" is (as I said) used for facts.
Gramatically, you have to build a subordinate clause after "wissen".
- "Ich kenne dieses Gefühl" means you already experienced this exact feeling.
- "Ich weiß, dass es dieses Gefühl gibt" means you know the feeling exists (because you read it on wikipedia or someone told you), but you might not have experienced this yet.
- "Ich kenne Hans Müller" - "I know Hans Müller (personally)"
- "Ich weiß, wer Hans Müller ist" - "I know who Hans Müller is"
Excellent explanation. Unfortunately, Duolingo sometimes gets this wrong. In this case, "kenne" is the right German verb, but in the English translation, Duo should accept "I am familiar with this feeling" as well as the more succinct "I know this feeling." I tried to report this, but did not get the automatic acknowledgement "Thanks for the report," so I don't know if they got the report. (I hope someone connected with German-English Duo is reading the comments.)