When comparing IRC vs Matrix, the Slant community recommends Matrix for most people. In the question“What are the best open source alternatives to Slack?” Matrix is ranked 1st while IRC is ranked 5th. 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 Open protocol
The IRC protocol is public and open, it is mature and stable, and has been implemented in numerous projects. This means that it can be used freely, for both open-source and commercial projects.
Pro Versatile
IRC offers choice. Choice in clients (mobile, desktop, web), in scripts those clients run, in servers and features those servers have. You can even set up your own server.
Pro Less distracting
Compared to other common solutions, IRC can be ignored when you are occupied doing something else. Most clients don't pester you with sounds or distracting visuals.
Pro Mentions
Every notable IRC client supports mentions - notifying you when your name is mentioned in the chat. The mentions in IRC are comparably better than Discord. Mentions used in Discord are easily abused; this doesn't happen in IRC. If someone's talking about you, you get a notification in IRC. This doesn't happen in something like Discord.

Pro Hundreds of avaliable clients
IRC has been around since 1988, so there is a ton of existing software to work with it. Everything from clients, to chat bots, to bouncers is already built.
Pro IRC networks are not centralized
You can choose your favorite server for an IRC network.
Pro No account necessary
No need to sign up for access, confirm your e-mail address. Just choose a nickname and you're done.
Pro User-friendly
IRC is user friendly if you use something like hexchat or kiwi webchat.
Pro DCC
Direct Client-to-Client allows users to share files in a similar way to torrenting.
Pro Great historic importance
Twitter hashtags came from IRC channel names.
Pro Upcoming IRCv3 will provide missing features
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 Complex and not user friendly
Con Assumes some level of prior knowledge
The features are not discoverable the way they are in other apps and services.
Con You have to "idle" to see what's going on
You will see a blank chat screen at first because there is no recent history kept on servers.
Con Can be addictive
Con Poor multimedia support
Sending anything besides text is not easy or intuitive and depends on the client.
Con Advanced functionality is client-dependant
Con Keys cannot be checked automatically
You cannot automatically check keys of your recipients. Only manually.
