At first glance, a puppet play feels like a good analogy for an average VN... I can't help to think of it primarily from the perspective of the writing process, but let's assume for now that the analogy works both ways. Writing a puppet play doesn't seem that distinct from writing a "real" one, but it forces you to compensate for additional limitations. Less in expressiveness of the characters and more in their capability to actually do things and present complex events to the audience. You have to rely more on narration (which you shouldn't overdo to not kill the pacing) and dialogue, as you're heavily limited in stuff that can be genuinely shown. Even the static backgrounds and many CGs are more like scenography that characters build the narrative over, populating them with people and action that would be too cumbersome to visually place there.
But then, it would be easy to argue this whole comparison is a fallacy, as you basically never get a VN read to you in full. Which makes it a primarily literary experience, but also a multimedia one that can neither be directly compared to reading a play nor to watching one. This makes comic books a more reasonable point of reference, but I feel like they operate with a completely opposite set of limitations. You can show a lot in them, despite the missing links between scenes that the reader has to fill with his imagination, but you can't tell much without bloating it with text. So, ultimately I'd land in a similar place as Clephas – the closest analogue to VN, from the perspective of the reader, is probably a Light Novel, with a bit of puppet play mixed in.