IRC > DISCORD

Discord's App doesn't allow installing to a custom directory.  IRC clients do.
Only one app serving Discord.  Many apps work with IRC.
IRC has a ton of third-party apps, as well as extensions for those apps, adding functionality.
IRC can work on a shoestring internet connection.
IRC uses plaintext format, making it easy to copy and paste elsewhere.
Discord allows emoji, making it difficult to copy and paste things in plaintext applications.
Discord uses various amounts of formatting that don't get copied as text.
Can't auto-save Discord logs.  This option is a common feature in IRC clients.
IRC, being plaintext, is often displayed in fixed-width font, which makes ASCII art much easier.  Discord requires special formatting to force fixed-width font.
And even then it doesn’t even offer a standard formatting for it.  You have to skip the first line or else stuff won’t line up.
When Discord goes down, all of Discord goes down.  When one IRC server goes down, the network usually has other servers still up and connected, and even if the entire network goes down, there are other networks to go to.
Joining a Discord server means you are automatically subscribed to all the channels.  You have to manually mute the channels in order to not get notifications from them.
And even then, you still show up in the user list for those channels.
Can't be present in only one channel on a server, e.g. idling a channel without making your presence known (unless the invite is public in which case you can make multiple Discord accounts and use multiple browsers I guess).
Can't have multiple connections to the same server in one program instance.  You can do this easily with IRC.
Wanna spam everyone with a notification?  @everyone.  Can't do this in IRC -- the more people you wanna spam-notify, the more work you're gonna have to do to type out each of their usernames.  That way you know exactly how many people are being spammed.  (Or you can make a bot do it for you, but still, it's less easy and less prone to abuse.)
Discord doesn't automatically refocus the text entry field after alt-tabbing away.  Have to type one character to make it focus the text entry field.
^ And not even that works, when your microphone is active.  You actually have to click the text entry field with your mouse cursor.
Discord requires an @ to start a mention; IRC doesn't.
Discord can't use tab to cycle through names.
Discord picks a name from any autocomplete list (e.g. an @ mention list) based on where your mouse cursor is, if it happens to be in the location of the menu.
Some IRC clients (e.g. X-chat and its derivatives) use Alt+digit to fast-switch between the first nine channels.  Discord can't do this, at least on its web version.  IRC clients also allow Ctrl+PgUp/PgDn to cycle through channels, which is faster than Ctrl+K and then going to stuff.
Can’t customize appearance of Discord to compress visual elements further than the “Compact” view.
Can’t save space by removing portraits from userlist.
Up + enter doesn’t re-enter the last line of text, but instead edits the last line of text.  This makes bot interactions and announcements more annoying.
Some bots’ lines are missing when used in certain browsers.
Some IRC clients have /anick and /ame user commands; Discord doesn’t.
You can just make an IRC channel public.  You don’t need to specifically post an invite for people, and specifically make it a non-expiring invite.
Closing Discord doesn’t mean that you’re gone.  People can still leave you messages, and people can expect that you’ll read up on them to see what you missed.  It’s more difficult to choose to tune out.
Can’t make a private channel for a subset of people on a server, such that other people on that same server are entirely unaware of that private channel.
^ Not only that, but IRC also lets anyone do this, not just server leadership.
Not forced to join certain IRC channels on first join (on most networks).  This is a common practice on Discord servers.

DISCORD > IRC

Discord has voice chat natively built-in.
Discord allows various rich formatting features such as emoji and link previews.
Discord allows various rich profile features such as avatars and non-standard characters in usernames.
Can automate user statuses with some games and other applications, natively (without third-party plugins).
Can control who’s in a server more easily, by using time-limited invites rather than just server/channel obscurity and channel passwords.  (Some private IRC networks use things like onjoin password commands for this.)
Discord saves chat logs so you can always see what chat content you missed.
Other people can leave you messages when you’re offline.  (However, if you nick is regged, IRC networks tend to have a memoserv feature for this already, though it may have limited inbox size.)
Can make a separate announcements channel where people can’t post but force people to be aware of its existence.
Can force channels to be displayed on new user join.  (Some private IRC networks do this.)
Can define user roles more flexibly – not stuck with owner/op/halfop/user or linear scale user levels, but can define custom per-channel permissions for users (and can also rename roles).
Easier to notify everyone using @everyone or @here.
You can talk about someone without pinging them, since mentions require an @.