as a "mere user" of the JS or API endpoints, should I think one is better than the other? To me it doesn't really matter if it's Supabase/Neon, Turso, or D1 powering it as long as... it works
With Turso you get an sqlite database behind a SDK or an (HTTP REST) API endpoint.
With Neon you get a postgresql database over a postgres wire format connection.
So if you have an already existing app speaking postgres to a database somewhere, Neon is a drop-in-replacement, while Turso would require adapting to their custom API.
If you are creating a new service, you might need/want to take advantage of e.g postgres extensions [0] for storing geographical data or pg_vector for similarity searches etc. Or you simply need more stringent serialisability promises than what libsql / turso can provide.
But if you just writing something new from scratch and have "simple" demands, I think something like Turso looks cool (and cheap!).
I'd probably care because of the ecosystem. There's a bunch of developer tooling and mature ORMs around postgres. Granted, those same libraries/frameworks probably work with sqlite too.
I guess it's just trust. I trust postgres more than I trust sqlite for building big apps.
One last point: I'd look at feature set of both offerings. I suspect the features of the postgres offering to be more conducive to scale.