My personal answer:
I've only ever played two- DDLC+ (my first), and Saya no Uta. I've also watched a few anime based on VNs (Clannad and Steins; Gate come to mind). To me, I suppose I just really liked the stories the tell- that's the only reason I played those two. I had read overviews of them, they sounded appealing (the psychological horror, mainly), so I played them.
I noticed mainly that what I really enjoyed the most was the stories being told and the OSTs. As a composer, the music of VNs have sincerely captured my interest. They're almost like wallpaper music- just complex enough to stand alone as great tracks, but not complex enough to distract you from reading.
I don't know. I guess I'm asking currently as I have a few more game ideas, and while I've tossed around the concepts in RPG Maker/GameMaker Studio, I'm starting to feel a real connection to the VN genre as a storyteller. (Plus, the reception from complete strangers and friends alike has been extremely encouraging). I guess I'd really just like to hear why others enjoy the genre if you have a spare moment or two! Thank you so much for your answer
(Also relevant, the Lemma Soft forums are perhaps the first forums I've ever actually WANTED to be a community member of- everyone is just so friendly, and I think that's another reason why I'm enjoying VNs so much).