About Matrix Matrix is an open protocol for decentralised, secure communications.
Matrix Manifesto We believe:
People should have full control over their own communication. People should not be locked into centralised communication silos, but instead be free to pick who they choose to host their communication without limiting who they can reach. The ability to converse securely and privately is a basic human right. Communication should be available to everyone as a free and open, unencumbered, standard and global network.
On Discord, you cannot host your own server, and you cannot use any third party clients (without the threat of being banned).
You can host your own Matrix server, either on physical hardware, or a generic virtual machine you can rent from any number of ISPs. There are over a dozen compatible third-party clients (though many lack full feature coverage).
In summary, Discord is strictly a service. Matrix is a tool you can apply however you see fit.