I was always a fan of TeamSpeak mainly due to us having one for so long. It was fairly easy to setup and customize when needed which is why I still have a 128 license. I am only weeks into discord and honestly like what I have seen so far. It definitely is nice that people don't have to buy an app to use discord.
As a computer scientist I tend to really like Discord simply because one can easily integrate server-side Discord bots.
Furthermore, the layouts of the Discord text-channels (regarding history and aesthetics) are simply more pleasing.
I can log into multiple Discord channels without the needing to open up multiple clients like in TeamSpeak (without using some sort of client-side mod)
Discord is beginning to support video messages
The hosting of one or more Discord servers is free
Discord's Android app is free
If I'm on the go and I only have a web browser, I can use the web app for Discord
Obviously TeamSpeak is more mature and they've nailed down the quality a bit better (but Discord is improving), but even then I've preferred low-latency programs like Mumble over TeamSpeak. That being said, I hang around my various TeamSpeak servers because more people my age tend to use it, but I think as a program Discord seems to be nicer.