Choose Neon when the database is often idle and you would rather pay $0.106 per CU-hour on Launch than a fixed size. Choose PlanetScale when you want a fixed monthly cluster price and you are ready to skip a free tier.
Neon
Serverless Postgres that can scale to zero
$0.106/CU-hour on Launch
Strengths
You pay for compute hours, not a always-on box
Branching is built in
Limits
A busy database does not stay on the free allowance