When comparing TeamSpeak 3 vs Discord, the Slant community recommends Discord for most people. In the question“What is the best voice chat for gaming?” Discord is ranked 2nd while TeamSpeak 3 is ranked 4th. The most important reason people chose Discord is:
Discord follows the same type of interface design popularised by Slack, which is extremely clean and attractive, and doesn't clutter the interface with unnecessary chrome and cruft.
Specs
Ranked in these QuestionsQuestion Ranking
Pros
Pro A lot of features
Pro Fine-grained administration options
Allows for complex administrative structures, with lots of roles and many other features related to server administration.
Pro Advanced permissions
It has an advanced system to manage channel/server permissions.
Pro Works on all main PC OS platforms
Available on Windows, Linux and OS X allows for a user to choose what best OS suits them and not have to worry if their game chat software will work.
Pro Supports audio and text chat
TeamSpeak allows for both voice and text chat. this adds flexibility, and can clarify information if somebody has a very poor connection, or a very thick accent. It also allows players without a mic to communicate with the other players.
Pro Widely used
Among most MMOs (massive multiplayer online games such as Guild Wars 2, or World of Warcraft) Teamspeak is widely used so it's likely other players already use it and won't need to download a new program. There are also many knowledge and help articles available online.
Pro Voice Activation Detection while using Push-To-Talk
Pro Echo reduction
Pro Free for non-commercial use
As long as one is not using TeamSpeak for commercial services the app is free to use. This is great as most users will not be using it for commercial purposes (as long as you aren't making money off it you don't need to pay for it) such as gaming.
Pro Unlimited nesting of channels
This allows for very granular organization.
Pro Sound level tuning
You can tune everyone's sound output levels to meet your own personal preference in inbound volume. So to your ears nobody is too loud or too quiet.
Pro Supports Encryption
Under "Edit Channel" --> "Advanced" --> theres a toggle button called "Voice Data Encrypted". Everything but file transfers is encrypted.
Pro Plugin support
A wide range of plugins are available for those that want extra functionality, ranging from soundboards and overlays, to peripheral and application hooks.
Pro Private hosting
You can host your own server.
Pro TeamSpeak Sync
Signing up for a MyTeamSpeak account will allow you to synchronize your bookmarks, identities, hotkeys, whisper lists and channel subscriptions across all devices. MyTeamSpeak features will include addons synchronization, addons update management, and more.
Pro Clean theme
Pro No privacy nightmare
Pro User friendly
Discord follows the same type of interface design popularised by Slack, which is extremely clean and attractive, and doesn't clutter the interface with unnecessary chrome and cruft.
Pro Free for unlimited users
Unlimited amount of users.
Pro Can be used in browser
Users can invite others to join a Discord channel via a web link, which means no installation is required to use the software - which is one of the big typical barriers to entry for social software.
Pro Good 1-click invite system
Discord allows inviting people to a specific channel with a link. Links can be set to expire. Low effort/passive invite mechanics.
Pro Easy setup
You can quickly send invites to people and they can join in using the web client without having to create an account or install the app.
Pro Has low-latency VoIP support
Pro Supports video and text communication
Behaves and looks like Slack for regular text channels, but with the ability to create voice channels as well
Pro Numerous controls for setting member roles.
Deny access, Deny write message, Allow read messages etc.
Pro Ability to create bots
Discord provides an API for creating custom bots.
Pro Open source bots and management tools
Discord has a wide variety of bots for things from airhorn noises to automated server management, to leveling systems for chat. The bot API is open and there are dedicated sites for finding bots.
Pro Game integration
Discord shows which game each user is currently playing, which among other things helps users to form PUGs with minimal hassle.
Pro Dark theme out of the box
Pro Familiar for Slack users
For those that are used to the way Slack (a similar chat client) works, Discord will be familiar in its use, which lowers the learning curve quite a bit for those that have switched or tried Slack before.
Pro Low effort invite mechanics
Pro Android app has push notifications
Pro Syntax highlighting and Markdown support
The support is limited but still exists. It's a killer feature for dev teams.
Pro DDoS mitigation
Discord prevents users from being DDoSed out of games.
Pro Targeting of roles
Targeting roles allows for quickly requesting someone from a group e.g. artist, moderator, etc.
Pro Ability to thread channels
Ability to create channel categories for channels to sit inside of.
Pro Friendly payment policy
Free account have no tangible restrictions.
Premium account is just a way to say "thank you" to developers. And it work for all servers of discord (not as in slack). And it can be purchased by one who wants. In contrast to Slask, where all active members of the server should be "premium".
Pro Language localization
Have localization for many language, including Russian.
Pro Does not allow NSFW related things to be posted unless in a NSFW channel
Cons
Con Mobile app costs money
TeamSpeak for mobile platforms costs $1.99 for Android devices in Google play store or $4.99 for devices that run iOS.
Con Requires to self-host, rent or use a public server
TeamSpeak cannot be used without a private server (meaning TeamSpeak themselves do not provide servers). This adds complexity to the setup and may simply be out of reach for many users. There are, however, several free public servers available, Teamspeak offers to rent one, and many games have dedicated servers available through their respective websites.
Con Outdated UI
The client itself feels really old, and even though there are custom huds, most of them don't feel modern.
Con Proprietary software
It's not free.
Con No video communication
Currently there is no video chat for the app. While not essential for in-game communication, outside of the game it would help to strengthen online relationships with your gaming pals.
Con Too easy to hack
Your server may become vulnerable if you give admin away and do not configure your server properly.
Keeping a record of backups is important, if you broke something while editing the configuration or your server has become compromised due to negligence. You will then be able to spin up a new instance.
Con Proprietary software
Discord is closed-source, meaning their code isn't available for inspection or reproduction
Con Based on Electron = eats a lot of ram
Con Some functions cut out and available only with "Nitro" subscription
Con Developer's previous company was sued for privacy violations, among other things
See:
http://classactionlawsuitsinthenews.com/class-action-lawsuit-complaints/openfeint-green-international-privacy-class-action-lawsuit-complaint/
https://www.ftc.gov/enforcement/cases-proceedings/closing-letters/openfeint-inc
Discord's privacy policy also states that while they might not sell your data, a company they get sold to will quite possibly attain said data and thus may be in "more" shady hands.

Con No self-hosting avaliable
Discord is wholly hosted by Discord (the company), you cannot run private instances, and all servers are hosted on their infrastructure. For most people, this won't really impact their usage.
Con Not a generic "team chat" tool
All the UI and the features are aimed specifically at communicating with your gaming team/clan.
Con Very unstable on Linux
Con Anti-privacy license terms
Con Can change "Now Playing" status to something inappropriate
