Choose MongoDB Atlas when the data is documents and an M0 cluster is enough to prototype. Choose Neon when you want Postgres, compute that can scale toward zero at $0.106 per CU-hour on Launch, and database branching.
MongoDB Atlas
Managed document database with a free shared tier
Free M0, then paid
Strengths
M0 is a real free cluster for prototypes
Dedicated tiers cover production
Limits
A real bill splits compute, storage, and backup
Document modeling is a poor fit for relational workloads