Both have advantages and disadvantages. All-in-all, I believe Discord offers some great out of the box features, setting up a server is stupidly simple. However, while TeamSpeak 3 is a PITA to set up sometimes, it has great addon hook support for games. One TS3 addon I can think of is ACRE for ARMA 3 which uses the plugin to simulate military radio communications and implements static, and dynamic signal loss, echo based upon the player's surroundings all through the TeamSpeak client.
tl;dr - Discord for general purpose chat, TS3 for anything more sophisticated
In my eyes teamspeak is a lot better. If you dont have a lot of ram on your computer and your gaming and the game uses alot of your ram... Then you have the internet open that shit just rip me apart !
Depending on what your needs are, both are good. Hopefully Discord will stay around for a long time and stay alive much longer than Ventrilo has. Teamspeak definitely has some features that Discord could use but by far, Discord is the easiest to use.