The killer feature for me is TeamSpeak's "raid leader". You can assign someone to be able to instantly mute everyone else on the call while holding a key. It seems very convenient in, well, raid settings. No idea why Discord doesn't have that.
[1] https://support.discord.com/hc/en-us/articles/360011876531-S...
0: https://web.archive.org/web/20181229093058/https://support.d...
We tried really really REALLY hard to make a shift to Discord[2]. We even tried writing bots that mimic some TS features. We ended up using Discord to include the more casual gamers from the guild, but most of us still use TeamSpeak, with a few people who're on both to help coordinate the less tech-savvy folk.
Discord always seemed to target a much broader "gaming" audience. It's simple, it does what it does in a mediocre way, but that's okay, because it's a casual product and nobody cares (evidently). You need to be a bit more tech-savvy to connect to set TS/Vent up, but if you're serious about multiplayer gaming, then it's worth the extra effort.
I'd (and a lot of my friends probably) use Discord fulltime if they started catering just a bit more to non-casual gamers too.
[1]: I wouldn't be surprised if Discord's aimed at small gaming groups and that the feature was designed with small/medium sized groups in mind. I'd guess the average room size on Discord probably falls within that range.
[2]: Most people had it already, so adoption wasn't the issue, it was the usability. However, the fact that most people already had it speaks volumes about Discord's conversion strategy/onboarding.