Supabase is an open-source platform that supports PostgreSQL databases, offering powerful features for complex queries and easy API generation, while Firebase is a closed-source service that provides NoSQL databases with flexibility but limitations on complex queries. Supabase is hosted on AWS and limited to cloud hosting, whereas Firebase is hosted on Google Cloud and offers serverless functions for on-premises hosting. Firebase benefits from a larger community and extensive documentation, while Supabase, despite its smaller user base, allows for modification of source code and transparency.