They're totally different in terms of the scope of the project they're meant to be used for.
FastAPI is much closer to Flask in that it's trivial to throw up a single file with a couple of routes, and you can use something like SQLite yourself for persistence to disk, or install a couple of libraries for authentication or such (not sure if that's built in by now).
Django, on the other hand, requires a multi-step process to even start a project and creates a dozen of files, most of which are boilerplate, before you can even see a "hello world" route.
But in exchange, you get not just a program which deals with routing and templates and status codes, but much more - a world-class ORM which integrates pretty much transparently with multiple data stores like Postgres or SQLite, an amazing dashboard out of the box which is really handy for sharing with non-techies, a very mature ecosystem, and perhaps most importantly, the "one right way to do things", which makes it a lot more effective for collaboration between a bunch of engineers. It's of course not infallible, has a learning curve, and comes with a good few footguns that get the uninitiated, but the upside is a real upside. Migrations alone might justify it - for all the grief of merge conflicts between migrations right before a code freeze, I can't imagine how much worse it would be to not have them.
So I wouldn't say there's significant advantages to using either - they're different tools for different use cases. If you want to get something up quick that isn't too complex, FastAPI is great for that. If you know you're eventually going to rewrite Django but worse, you may as well use the real thing.