I've had both (and still have a Tutanota account). They are quite different. The main advantage of Tutanota is that is has zero-knowledge encryption (i.e. once the mails are stored even Tuta can't read or scan them, and even if others got access e.g. by hacking, law enforcement request or a bug, your emails would still be safe), as well as end-to-end encryption when communicating with other Tuta users.
Fastmail has no such thing, so you have to rely on their promises when it comes to privacy, and in case of a breach your emails could be exposed. Otherwise they are a very good email provider IMO, and one of the few that support push email with the stock iOS Mail app.
This comes with some disadvantages too: Tuta cannot be used with standard mail clients. Their mobile app isn't all that great (it's basically a web app and some things are quite slow), and you cannot use it with a standard desktop client (they have no equivalent to Protonmail's Bridge). They also currently don't have a way to import/export mails in bulk, so migration to/from Tuta is an issue (but they are supposedly working on that).
But their prices are actually really good. For $1/month you get aliases, custom domain support and 1GB of storage. Fastmail starts at, what, $3/month?