I have both set up right now.
Things I like better about KeePass:
KeePass doesn't use the cloud, you don't have to worry about the server getting compromised or going down because there's nothing public-facing to hack. You always know where your password database is.
KeePass lets you encrypt the database with not only the master password but also using the challenge-response from a YubiKey. That means every time you save your DB the encryption key is rotated and the DB is actually encrypted by two authentication factors.
While both can add custom fields to an entry, I like that KeePass has the option to set fields as protected so their contents are hidden like the passwords.
Things I like better about VaultWarden:
Convenience.
You can log in to your VaultWarden account on any device from the browser. KeePass requires some software to access the DB.
The VaultWarden companion software is just better. It just does autofill better. KeePassXC/DX work well but just not as well as the BitWarden software.
Other thoughts:
Syncing passwords between devices with KeePass requires 3rd party software like SyncThing. If you break/lose/etc your VaultWarden server you could lose all your passwords with it.
Always make/test backups.
Shouldn't that be all the versions?
Why would a password manager app that uses a local database need to phone home?