Both have docker based setups and getting them up and running is documented clearly enough.

The biggest difference right off the bat (ignoring PosgtreSQL vs MariaDB) is that Supabase selfhosted just gives you default project and that's it. If you want to host several projects you either spin up new instance or make use of table prefixes. With Appwrite you can create new projects that are independent of each other, including auth. You can have organizations -> projects -> databases granularity. For sideprojects that's a huge win for me.

Supabase gives you a way to run raw SQL on your tables via web interface which I really like, it gives a better way to gain insights into your data without wrestling with client side queries which may or may not do what you need. Appwrite doesn't have that, you have to make do with their queries.

Documentation is fairly fragmented in both, code snippets and examples have not been updated for changes, some new features are only discoverable via search but not via documentation menu e.g. Appwrite's SSR documentation. Most complete documentation in both seems to be for Javascript. I managed to get auth working, but it may require some head scratching, especially if you want to use something different than their default examples.

Appwrite gives you shell scripts for backup/restore right there in the documentation, Supabase does not. It's not anything fancy, but it's nice to have. There are no ways to generate backup from Supabase web interface either so you have to write your own scripts.

Appwrite gives you basic dashboard of stats for your databases and simple charts for traffic and requests, Supabase selfhosted lacks that. Managed version has reports so they're clearly withholding this in purpose.

The thing that made me not want to go with selfhosted Supabase in the end though was constant CPU usage when idling. Some usage is expected, but even on my workstation the load stays around 5-8% and is frequently peaking up to 12% or more. In contrast Appwrite stays between 0.6-3%. On my VPS idle peaks are up to 25% which is wild. The culprit seems to be Supabase analytics and health check which logs agressively and spams console several times per second. IIRC there was open issue about it but the response was "maybe disable health-checks then". This coupled with just having one single project makes it not suitable for self hosting for me. Haven't used functions or realtime in neither so can't comment on that. Overall Appwrite's self hosted solution is more polished, but YMMV.