Feature-wise, pretty much nothing, it's just targeting a different set of users. Slack started as a solution for companies and pretty much stayed there, Discord started as a solution for gamers and expanded from there.

Slack is Discord for adults, but adults are never the ones who make or break social media. It's younger people who tend to be more invested in the platform they use. It's not a rational choice (nor was Facebook for my generation), but it got kickstarted out of necessity (as a substitute to the limited chat options within games) and right now, it's the convenience that drives it further. Why switch over when, unlike Facebook or Slack, everyone you know is already using Discord?