Why should I care ? Honest question.
Whereas in ATproto you have to find a place to register an account, say a domain for yourself or use someone else's domain, in NOSTR creating an account is nothing more than a public/private key that is generated on the fly.
Then is the storage of information. Whereas in ATproto information is stored in plenty of different formats and specific folder configurations which are easy to miss, in NOSTR is all JSON from a single base template file.
Third is the absence of global moderation and islands. Whereas in ATproto only a few (I think 3 main providers) exist, you can be kicked out from them and effectively stop reaching your audience. In NOSTR if a relay stops receiving your texts it is a bummer but there are hundreds of other volunteer-based servers to where you can send your texts, so it is very difficult to block people from reading texts of the authors they are interested.
Simplicity is the key. NOSTR is the first and only network that I've seen so far where content/authors are truly disconnected from the servers hosting their data.
OK so Nostr is for people that look for pure decentralisation ? This decentralisation so complicated that no general public person will ever connect ?
Where owning lemonde.fr doesn't distinguish you from a user pretending to be the biggest national French newspaper ?
Le Monde adds a text file at https://lemonde.fr/.well-known/nostr.json with the npubs (NOSTR public keys) of every person that they certify as trusted.
So then you go around using "maelito@lemonde.fr" and anyone can find your public key on the official Le Monde site, verify that you are indeed someone associated to that domain.
The reason people might care is taking ownership of how they filter and curate their feed. Instead of outsourcing that to some algorithm owned by a company, having a federated network allows you to pick and choose or even implement your own filtering if you self host. It also gets advertising as the primary revenue mechanism out of the way and that removes a conflict of interest between what users want to see and what advertisers want them to see.
Of course the drooling masses on Tik Tok, Instagram, etc. seem to be perfectly happy to have their endocrine system manipulated by self tuning algorithms. But there are some legitimate concerns about whether that's actually healthy for them. Federated social networks provide an alternative to this.
Extensible, resilient communications without the threat of a central third party silencing you is quite appealing.