The answer is, like a lot of things in life, it depends. Both are password managers, but they’re different and it will depend on what problem you’re trying to solve and what level of tech savvy and complexity you feel comfortable with. Strongbox leaves a lot of options up to you, Bitwarden reduces some complexity by adding some costs which might be acceptable for you.
One of the key differences is the requirement for a server. Strongbox does not require a server. Bitwarden does. Strongbox can be used completely offline, it uses a simple single file (encrypted really well) that you manage yourself.
You can put it on a pen drive and carry it around in your pocket. You can leave it on your mobile device or sync it to and from a public cloud drive (e.g. Dropbox, Google Drive). You can store it on your own private server (via SFTP, WebDAV).
Bitwarden requires a server to host, manage and provide your passwords. You can choose to have someone host this for you (Bitwarden offer this) or you can host yourself. They also provide a fairly wide range of clients (iOS, Android, Browser Plugins, Desktop apps) to interact with the server holding your data. There are some advantages to the client/server model, synchronization is made easier, the server is the single source of truth, and you don’t need to worry about forgetting your pen drive.
Strongbox provides ultimate control over your most prized secrets through a simple encrypted file that you own and manage completely.
It is also a advantage not to have to depend on a server being available, secured and and maintained.
However there is a cost to this independence. Don’t lose that file and keep regular backups!
Another advantage of Strongbox is that the database (the file mentioned above) is in a well known format (KeePass format is the default) that can be opened by many other clients across lots of platforms, so your data is not locked in to Strongbox. You can use KeePass on Windows, KeePassXC, KeePass2Android etc with the exact same file.
Bitwarden is a more managed solution and can be a bit easier for your less tech savvy user. In this way it is very similar to fully managed solutions like LastPass etc.