The similarities are that you can use them both to set up federated voip calls and instant messages. SIP is still the standard way for doing interoperable VoIP on the net.
The differences are huge:
* SIP is geared up for initiating sessions (hence the name), whilst Matrix is for synchronising data. SIP is heavily modelled after the requirements of the old PSTN phone network.
* It's quite an old and fragmented protocol and interop can be a pain, especially between IETF and 3GPP style variants
* It has many competing messaging systems layered on it (SIMPLE, MSRP, etc). You have to try very hard to extract a modern chat experience out of them - eg synchronised history, let alone end to end crypto.
* Matrix is HTTP; SIP is its own dedicated UDP-or-TCP transport.
Superficially it's HTTP-like, but just enough to give you a false sense of familiarity. (For context, we spent ~10 years writing commercial SIP stuff before ditching it all in favour of Matrix).
You can just about reuse the request/response header & mime parsing stuff (most of which is as much SMTP as HTTP), but the actual stack is completely different - supporting transports on unreliable transports like UDP via three-way handshakes (INVITE -> 200 OK -> ACK), both client->server and server->transactions, different kinds of transactions, registration methods, multiple responses, and all manner of voodoo when it gets to handling all the various types of routing headers. There are two entirely separate abstractions of the stack (ignoring the parsing layer) - the transaction layer (handling those INVITE->200->ACK things that look a bit like HTTP requests but aren't), and then the dialog engine on top that actually the lifecycle of a session (e.g. a phone call).
Random anecdote: a few years a large European telco launched a chat/comms service based on RCS (a GSMA set of standards built on SIP, IMS, MSRP etc). We sniffed the wire to see what happened if you sent the three letters "Hi!" from one client to another. Suffice it to say that it involved 85 round trips and and 54 kilobytes of data (all of which was plaintext, comically) - and this ignored the initial SIP registration. In order to send that message it set up about 3 different sessions (one to send the message as SIP MESSAGE, another to do it again by MSRP (including all the MSRP handshakes) in order to include a typing notification, another to receive the read receipt, etc).
This is why the world needs standards like XMPP and Matrix as a more internetty and less telcoy alternative ;)
Matrix does voip too (Console and Vector implement it using WebRTC as the transport) but it's not as mature as SIP. It's fine for casual use but you wouldn't want to run a telco off it yet :) That said, it's roughly 20x easier to develop on than SIP ;)
The differences are huge: * SIP is geared up for initiating sessions (hence the name), whilst Matrix is for synchronising data. SIP is heavily modelled after the requirements of the old PSTN phone network. * It's quite an old and fragmented protocol and interop can be a pain, especially between IETF and 3GPP style variants * It has many competing messaging systems layered on it (SIMPLE, MSRP, etc). You have to try very hard to extract a modern chat experience out of them - eg synchronised history, let alone end to end crypto. * Matrix is HTTP; SIP is its own dedicated UDP-or-TCP transport.