Relay/instance admins can choose which content goes through their relay on either platform
On nostr, your DMs are encrypted. In Mastodon, the admin of the sender and receiver can read them, as can anybody else who breaks into their server
On nostr, a relay admin can control what goes through their relay, but they can’t stop you from following/DMing/being followed by whoever you want since you are typically connected to multiple relays at once. As long as one relay allows it, signal flows. Nostr provides the best of both worlds: moderated “public squares” according to your moderation preferences, autonomy to follow/dm/be followed by anybody you want (assuming that individual user hasn’t blocked you).
On mastodon, your identity is tied to your instance. If your instance goes down, you lose your follow/followee list, DMs, etc. On Nostr, it’s not, so this doesn’t happen. Mastodon provides some functionality to migrate identity between instances but it’s clunky and generally requires to have some form of advanced notice.
Both have all the same functions as twitter: tweet, reply, re-tweet, DM, like, etc.
Nostr vs Mastodon on Privacy & Autonomy:
Why I think nostr will win https://lemmy.ml/post/11570081
Nostr might be more private and autonomous, but it’s full of crypto shams.