When comparing Discord vs Matrix, the Slant community recommends Matrix for most people. In the question“What is the best chat software?” Matrix is ranked 1st while Discord is ranked 10th. The most important reason people chose Matrix is:
Matrix is an open standard, defining simple HTTP APIs so that devs can easily write their own clients, bots, bridges or servers. You're not locked into a specific set of implementations.
Specs
Ranked in these QuestionsQuestion Ranking
Pros
Pro User friendly
Discord follows the same type of interface design popularised by Slack, which is extremely clean and attractive, and doesn't clutter the interface with unnecessary chrome and cruft.
Pro Free for unlimited users
Unlimited amount of users.
Pro Can be used in browser
Users can invite others to join a Discord channel via a web link, which means no installation is required to use the software - which is one of the big typical barriers to entry for social software.
Pro Good 1-click invite system
Discord allows inviting people to a specific channel with a link. Links can be set to expire. Low effort/passive invite mechanics.
Pro Easy setup
You can quickly send invites to people and they can join in using the web client without having to create an account or install the app.
Pro Has low-latency VoIP support
Pro Supports video and text communication
Behaves and looks like Slack for regular text channels, but with the ability to create voice channels as well
Pro Numerous controls for setting member roles.
Deny access, Deny write message, Allow read messages etc.
Pro Ability to create bots
Discord provides an API for creating custom bots.
Pro Open source bots and management tools
Discord has a wide variety of bots for things from airhorn noises to automated server management, to leveling systems for chat. The bot API is open and there are dedicated sites for finding bots.
Pro Game integration
Discord shows which game each user is currently playing, which among other things helps users to form PUGs with minimal hassle.
Pro Dark theme out of the box
Pro Familiar for Slack users
For those that are used to the way Slack (a similar chat client) works, Discord will be familiar in its use, which lowers the learning curve quite a bit for those that have switched or tried Slack before.
Pro Low effort invite mechanics
Pro Android app has push notifications
Pro Syntax highlighting and Markdown support
The support is limited but still exists. It's a killer feature for dev teams.
Pro DDoS mitigation
Discord prevents users from being DDoSed out of games.
Pro Targeting of roles
Targeting roles allows for quickly requesting someone from a group e.g. artist, moderator, etc.
Pro Ability to thread channels
Ability to create channel categories for channels to sit inside of.
Pro Friendly payment policy
Free account have no tangible restrictions.
Premium account is just a way to say "thank you" to developers. And it work for all servers of discord (not as in slack). And it can be purchased by one who wants. In contrast to Slask, where all active members of the server should be "premium".
Pro Language localization
Have localization for many language, including Russian.
Pro Does not allow NSFW related things to be posted unless in a NSFW channel
Pro Built on an open standard
Matrix is an open standard, defining simple HTTP APIs so that devs can easily write their own clients, bots, bridges or servers. You're not locked into a specific set of implementations.
Pro Bridges other networks into a single decentralised network
Matrix has bridges to IRC (freenode, moznet, oftc, snoonet etc), Slack, Gitter, Rocket.Chat, XMPP, SMS, SIP and others. The point is to 'matrix' all the different networks out there into one single decentralised network.
Pro Has an easy to use client called Riot
Riot.im is the easiest way to use Matrix, with great clients for Web, iOS and Android (and Fdroid).
Pro Does not require a centralized server to establish a connection between two users
Matrix is decentralized, there's no one central point that the information goes through and so no once central point of failure or control.
Pro Matrix prioritizes direct messaging with people the same as Slack-style groups
Matrix aims to "provide an analogous ecosystem to email - one where you can communicate with pretty much anyone, without caring what app or server they are using" using a neutral identity system.
Pro Supports different kinds of communication
Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling (voice and video) and Internet of Things communication.
Pro Has an app store for 3rd party integrations & bots
Riot.im includes an app store with integrations for Github, JIRA, Jenkins, Giphy etc - and anyone can add more via Matrix.
Pro Offers choice of clients
Which can be found here.
Pro Maintains full conversation history
Pro Has an active community behind it
Pro Can be integrated with existing communication services
Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling and Internet of Things communication and allows cross-communication between those services. Meaning one person could be using IRC and another Slack for IMs, or one person could be using Skype and another Google Hangouts for videoconferencing.
Pro End-to-end encryption
Matrix features end-to-end encrypted chats which are syncronized accross all your devices.
E2E is implemented in the matrix-js-sdk library and Riot.im client.
Cons
Con Proprietary software
Discord is closed-source, meaning their code isn't available for inspection or reproduction
Con Based on Electron = eats a lot of ram
Con Some functions cut out and available only with "Nitro" subscription
Con Developer's previous company was sued for privacy violations, among other things
See:
http://classactionlawsuitsinthenews.com/class-action-lawsuit-complaints/openfeint-green-international-privacy-class-action-lawsuit-complaint/
https://www.ftc.gov/enforcement/cases-proceedings/closing-letters/openfeint-inc
Discord's privacy policy also states that while they might not sell your data, a company they get sold to will quite possibly attain said data and thus may be in "more" shady hands.

Con No self-hosting avaliable
Discord is wholly hosted by Discord (the company), you cannot run private instances, and all servers are hosted on their infrastructure. For most people, this won't really impact their usage.
Con Not a generic "team chat" tool
All the UI and the features are aimed specifically at communicating with your gaming team/clan.
Con Very unstable on Linux
Con Anti-privacy license terms
Con Can change "Now Playing" status to something inappropriate
Con Keys cannot be checked automatically
You cannot automatically check keys of your recipients. Only manually.
