Matrix is an open standard protocol (https://matrix.org/docs/spec) and an open network like the Web, but for realtime comms. Anyone can spin up a server and start participating in the network, and in Matrix the conversations are replicated over all the servers whose users participate in that conversation, so there is no single point of failure or control: the conversations are owned by the users. All the implementations we release as matrix.org are Apache licensed FOSS.

Now, Keybase is cool - particularly how it's effectively a key management system which drives a collaboration tool. We hoped originally they would provide it as a decentralised identity management system that we could hook into Matrix to replace our identity service. But in practice it seems to be centralised, and whilst the client is FOSS (https://github.com/keybase/client) the server isn't. So, it's more an E2EE-capable centralised collaboration tool built on a cool key management system. I believe their E2EE is based on shared keys rather than a cryptographic ratchet (Matrix uses inspired-by-Signal Double Ratchet with extensions), which is a simpler approach, but doesn't give you the option of PFS. I'm also not sure if their crypto has been audited (whereas we have a slightly ageing audit of Matrix's implementation over at https://matrix.org/blog/2016/11/21/matrixs-olm-end-to-end-en...).

[disclaimer: I'm project lead of Matrix and the author of the OP]