However, architecturally they really couldn't be more different:
* Matrix's main data primitive is synchronising conversation history within a room - not message passing. In fact, there is no way to just 'send a user a message' in Matrix: instead, you can only synchronise your copy of a room's state and history with someone else's copy of it.
* Matrix rooms are replicated equally over all participating servers - there is no focal point as there is in a XMPP MUC.
* Matrix provides a single monolithic (beta) spec, which compliant clients have to implement (for a given class of clients). There are no optional features or competing extension proposals for a given feature for a given class of client (e.g. 'desktop messenger'), to try to avoid fragmentation.
* Matrix's baseline transport and encoding is super-simple-stupid HTTP+JSON, but with room for folks to propose superior transports & encodings as we see fit. So far there's been WS+JSON (which nobody seems to use, as it's not that massive an improvement over HTTP+JSON), and this year there's a GSoC project to look at MQTT-and-similar as an alternative.