The fact that you don't need to install or configure any client (or even have an account) to bring someone in is the biggest single thing.
People come for the easy invites and stay for the (slightly better than) good enough chat, voice, and community features.
I'm sure there's plenty of guilds around that are still happy with their existing setup and haven't bothered switching, but my experience is that many have switched, and new guilds are basically only picking Discord.
I can see it pulling away from pretty much everything except those groups that have spent lots of time on crazy teamspeak setups with lots of ranks, different overlapping groups and so on.
Is TeamSpeak an open API? Shouldn't be difficult at all to write an import tool to import a TeamSpeak server's group structure to equivalent Discord roles.
Ultimately, I don't think this is as much an obstacle to long term success of Discord. These kind of servers tend to come around because running TS/Mumble servers require expertise certain groups don't have, so groups tend to colocate over some overlap in membership, and may still be colocated over inertia even if the overlap goes away.
The example I'm thinking of is a server I'm still on which is nominally owned by a battlefield clan, but also has a ffxiv guild, wildstar guild, eve corp and minecraft server chat on it. Each of these are effectively seperate entities and need to manage their own permissions, and if they were setting up fresh now, they'd all be seperate discord servers. The battlefield guys have some crazy rank structure based on time on the server or some such, the ffxiv group just has admin or not, and the minecraft group is a yes/no access perm.
The colocation is particularly not ideal as the battlefield clan and minecraft server tend to bring occasional DDoS attempts on the server, and there's not really much overlap in membership nowadays.
From the DM point of view he can still send us pictures, it is free and we don't really use maps so don't need Roll20 (or equivalent). Also direct messaging players secret messages is really easy and handy.
Discord's rise helped by Microsoft's seeming neglect and confusing platform direction of Skype in recent years, even if Discord still doesn't have video chat. For messaging among people who know each other by name, Facebook's offerings are dominant, but for pseudonymous chat Skype remained strong until Microsoft vacillated between making Skype Windows' native messenger or a strong multiplatform freemium network that isn't plagued by an aging UI and the invasion of advertising. Discord won many users by organic word-of-mouth referrals from people who used to use Skype.
As for gaming voice chat, Discord is not as sophisticated as other offerings (edit: maybe that's the wrong wording; let's go instead with "not as configurable"), but offers a smoother out-of-the-box experience.