AT proto has some significant similarities to Matrix:

* Both are work by self-authenticating git-style replication of Merkle trees/DAGs

* Both define strict data schemas for extensible sets of events (Matrix uses JSON schema - https://github.com/matrix-org/matrix-spec/tree/main/data/eve... and OpenAPI; AT uses Lexicons)

* Both use HTTPS for client-server and server-server traffic by default.

* Both are focused on decentralised composable reputation - e.g. https://matrix.org/blog/2020/10/19/combating-abuse-in-matrix... on the Matrix side, or https://paulfrazee.medium.com/the-anti-parler-principles-for... on the bluesky side, etc.

* Both are designed as big-world communication networks. You don't have the server balkanisation that affects ActivityPub.

* Both eschew cryptocurrency systems and incentives.

* Both have names which everyone complains about being hard to google, despite "AT protocol" and "Matrix protocol" or "Matrix.org" being trivial to search for :P

There are some significant differences too:

* Matrix aspires to be the secure communication layer for the open web.

* AT aspires (i think) to be an open decentralised social networking protocol for the internet.

* AT has portable identity by default. We've been working on this on Matrix (e.g. MSC1228 - https://github.com/matrix-org/matrix-spec-proposals/pull/122... and MSC2787 - https://github.com/matrix-org/matrix-spec-proposals/blob/nei...) and have a new MSC (and implementation on Dendrite) in progress right now which combines the best bits of MSC1228 & MSC2787 into something concrete, at last. In fact the proto-MSC is due to emerge today. EDIT: and here it is: https://github.com/matrix-org/matrix-spec-proposals/blob/keg...

* AT is proposing a asymmetrical federation architecture where user data is stored on Personal Data Servers (PDS), but indexing/fan-out/etc is done by Big Graph Servers (BGS). Matrix is symmetrical and by default federates full-mesh between all servers participating in a conversation, which on one hand is arguably better from a self-sovereignty and resilience perspective - but empirically has created headaches where an underpowered server joins some massive public chatroom and then melts. Matrix has improved this by steady optimisation of both protocol and implementation (i.e. adding lazy loading everywhere - e.g. https://matrix-org.github.io/synapse/latest/development/syna...), but formalising an asymmetrical architecture is an interesting different approach :)

* AT is (today) focused on for public conversations (e.g. prioritising big-world search and indexing etc), whereas Matrix focuses both on private and public communication - whether that's public chatrooms with 100K users over 10K servers, or private encrypted group conversations. For instance, one of Matrix's big novelties is decentralised access control without finality (https://matrix.org/blog/2020/06/16/matrix-decomposition-an-i...) in order to enforce access control for private conversations.

* Matrix also provides end-to-end encryption for private conversations by default, today via Double Ratchet (Olm/Megolm) and in the nearish future MLS (https://arewemlsyet.com). We're also starting to work on post quantum crypto.

* Matrix is obviously ~7 years older, and has many more use cases fleshed out - whether that's native VoIP/Video a la Element Call (https://element.io/blog/introducing-native-matrix-voip-with-...) or virtual worlds like Third Room (https://thirdroom.io) or shared whiteboarding (https://github.com/toger5/TheBoard) etc.

* AT's lexicon approach looks to be a more modular to extend the protocol than Matrix's extensible event schemas - in that AT lexicons include both RPC definitions as well as the schemas for the underlying datatypes, whereas in Matrix the OpenAPI evolves separately to the message schemas.

* AT uses IPLD; Matrix uses Canonical JSON (for now)

* Matrix is perhaps more sophisticated on auth, in that we're switching to OpenID Connect for all authentication (and so get things like passkeys and MFA for free): https://areweoidcyet.com

* Matrix has an open governance model with >50% of spec proposals coming from the wider community these days: https://spec.matrix.org/proposals

* AT has done a much better job of getting mainstream uptake so far, perhaps thanks to building a flagship app from day one (before even finishing or opening up the protocol) - whereas Element coming relatively late to the picture has meant that Element development has been constantly slowed by dealing with existing protocol considerations (and even then we've had constant complaints about Element being too influential in driving Matrix development).

* AT backs up all your personal data on your client (space allowing), to aid portability, whereas Matrix is typically thin-client.

* Architecturally, Matrix is increasingly experimenting with a hybrid P2P model (https://arewep2pyet.com) as our long-term solution - which effectively would end up with all your data being synced to your client. I'd assume bluesky is consciously avoiding P2P having been overextended on previous adventures with DAT/hypercore: https://github.com/beakerbrowser/beaker/blob/master/archive-.... Whereas we're playing the long game to slowly converge on P2P, even if that means building our own overlay networks etc: https://github.com/matrix-org/pinecone

I'm sure there are a bunch of other differences, but these are the ones which pop to the top of my head, plus I'm far from an expert in AT protocol.

It's worth noting that in the early days of bluesky, the Matrix team built out Cerulean (https://matrix.org/blog/2020/12/18/introducing-cerulean) as a demonstration to the bluesky team of how you could build big-world microblogging on top of Matrix, and that Matrix is not just for chat. We demoed it to Jack and Parag, but they opted to fund something entirely new in the form of AT proto. I'm guessing that the factors that went into this were: a) wanting to be able to optimise the architecture purely for social networking (although it's ironic that ATproto has ended up pretty generic too, similar to Matrix), b) wanting to be able to control the strategy and not have to follow Matrix's open governance model, c) wanting to create something new :)

From the Matrix side; we keep in touch with the bluesky team and wish them the best, and it's super depressing to see folks from ActivityPub and Nostr throwing their toys in this manner. It reminds me of the unpleasant behaviour we see from certain XMPP folks who resent the existence of Matrix (e.g. https://news.ycombinator.com/item?id=35874291). The reality is that the 'enemy' here, if anyone, are the centralised communication/social platforms - not other decentralisation projects. And even the centralised platforms have the option of seeing the light and becoming decentralised one day if we play our parts well.

What would be really cool, from my perspective, would be if Matrix ended up being able to help out with the private communication use cases for AT proto - as we obviously have a tonne of prior art now for efficient & audited E2EE private comms and decentralised access control. Moreover, I /think/ the lexicon approach in AT proto could let Matrix itself be expressed as an AT proto lexicon - providing interop with existing Matrix rooms (at least semantically), and supporting existing Matrix clients/SDKs, while using AT proto's ID model and storing data in PDSes etc. Coincidentally, this matches work we've been doing on the Matrix side as part of the MIMI IETF working group to figure out how to layer Matrix on top of other existing protocols: e.g. https://datatracker.ietf.org/doc/draft-ralston-mimi-matrix-t... and https://datatracker.ietf.org/doc/draft-ralston-mimi-matrix-m... - and if I had infinite time right now I'd certainly be trying to map Matrix's CS & SS APIs onto an AT proto lexicon to see what it looks like.

TL;DR: I think AT proto is cool, and I wish that open projects saw each other as fellow travellers rather than competitors.