When comparing Viber vs Discord, the Slant community recommends Discord for most people. In the question“What is the best team chat software?” Discord is ranked 8th while Viber is ranked 38th. 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 Desktop apps allow texting, calling and enables transferring calls between devices
Viber has native apps for both Windows and Mac that allow you to send and receive messages from your desktop. You can also use the desktop to make and receive calls. Synchronizes between all devices you are using Viber on. Allows transferring calls between devices.
Pro Supports voice calls
Viber has really well implemented voice tech. It has comparable quality to Skype and you can even tranfer calls from your phone to your desktop seamlessly.
Pro Completely free
Viber uses data connection and the app is free.
Pro Clean interface
The interface is simple, functional and intuitive.
Pro See who of your contacts is using Viber quickly
Viber is tied to your phone number so it can quickly see who else in your contacts list is using the application.
Pro Android Wear support
Viber includes Android Wear support on Android 4.3 or higher and Viber 5.3 or higher.
Pro Same protocol as used in Signal
Viber's protocol uses the same concepts of the "double ratchet" protocol used in Open Whisper Systems Signal application.
Pro Public chats
It's possible to be a fly on the wall on other group or one-to-one chats that have been opened up to the public. These public chats are often promotional chats with celebrities.
Pro Supports video calling over WiFi and data.
Can call friends using voice or video options.
Pro Combines cellular and data messaging
This has an advantage in that if you text someone who uses Viber it will be free, if not it will just revert to SMS. This means you can replace your standard messaging app with Viber as well.
Pro Keys only on device
Starting with Viber 6.0 the encryption keys are stored only on the clients themselves, not even Viber itself has access to them.
Pro No need to create an account or add contacts
One of the best things about Viber is how it handles setup and sending messages. Viber uses your existing number as your ID, so you don't need to create a separate Viber account.
Pro Supports 200 person group chat
Viber allows creating groups of up to 200 people on the Android version of Viber. It also has a cool "smart notifications" feature to prevent being overwhelmed with notifications if a lot of people are active in a chat group.
Pro Insight and well documented encryption
Viber has made a documentation available, explaining in great details their encryption system and giving good insight on how it works.
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 The company evades U. S. tax
The company uses the common practice of tax haven addresses in order to avoid paying US taxes despite being registered in Delaware.
Con Proprietary Software
Not Open Source.
Con No web client
You cannot login from the web to send and receive messages.
Con Constantly on
Need apps like Greenify to stop it from being constantly on and active, even if you don't use it.
Con Constant reminders of messages
If you don't check your messages (say your in a group chat and simply don't check all of them), it will flood your notifications.
Con Doesn't sync well
Sync awfully between Phone and PC clients.
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
