When comparing Hangouts 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 Hangouts is ranked 15th. 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 Save and sync your chats across multiple devices
Start chatting on your phone, then on your computer, then on your tablet seamlessly!
Pro No need to create a separate account if you use any other Google service
Hangouts uses Google account credentials. If you use any other Google service, then you already have an account set up.

Pro Group video support
If you have signed in your Google account, Hangouts allows making video calls with up to 10 people or one-on-one. There are business, education and government solutions that increase the participant count to for video calls to 15 as well as offers support for those accounts.
Additionally, you can switch between using either the front-facing or rear-facing camera on your phone or tablet.

Pro Free and cross-platform
Google offers all features of Hangouts completely free on iOS, Android and web with quick and reliable sync across all platforms. It's also tightly integrated in other Google services - it can be accessed from services such as Gmail, Google+, Chrome.
While the standard Hangouts offering is completely free, there are business, education and government solutions that increase the participant count to for video calls to 15 and offers support that may cost money.
Pro Chrome browser extension
Use the extension on any computer you can run Chrome browser to text from your computer or tablet!

Pro Excellent read receipts in group chat
It's possible to know what each and every participant of the group has read. Their icons are displayed below the last message.

Pro Can share animated GIFs
If you have gifs in your mobile phone, you can share them in the conversation and your partner will be able to see it animated in the chat.
Pro Google Voice integration
Google Voice is a service that enables making and receiving free phone calls within US. Google Voice forwards incoming calls to all connected devices and allows placing calls from any of the connected devices, but they will show as coming from the Google Voice number.
Hangouts completely integrates Google Voice, allowing you to place calls and receive texts and voicemail from all of your devices. The Hangouts app for Android also includes a needed dialer for making calls with the users Google Voice number.
Pro Free calls to US and Canada from anywhere in the world
Hangouts app for Android can be extended with an optional dialer for making calls. Most calls to US and Canada are free and international calls have cheap rates.
A Google Voice account will be needed in order have a number that allows these calls to be made.
Pro Advanced status sharing
Hangouts can inform about the status of other users including what device they are currently using, whether they are in the middle of a call and when were they last seen. This allows making sure that you're not disturbing someone when initiating a conversation. It's possible to granularly turn off status sharing.
Pro Video tools
Hangouts can add different effects and overlays to video calls that can be viewed on your device. Unfortunately, they can't be set from your phone.
Pro Seamless integration with OK Google for voice control and typing
Hangouts can be used completely hands free, even if the app is not open and the phone is locked: "OK Google, send a Hangouts message to Mary".
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 Transitioning to corporate orientation
Google is turning Hangouts in to a Slack competitor. It's unclear what the future of Hangouts is, but Google has said the consumer Hangouts we now have 'isn't going anywhere.'
Con Lack of end to end encryption
Communications are encrypted between client and Google servers but not from sending client to receiving client. This means that there is still a possibility for someone other than the receiver to read your message if they have access to internal Google servers. But it allows also lightweight chat history search, spam protection and other features.
Con Is owned by Google
Google is a cooperation that earns money from collecting data and breaching privacy.
Con Poor interface
The app tends to lag, the design can be confusing and considering that another app has to be installed that supplies another icon in the home screen in order to get a dialer for calls in Hangouts, overall it is very confusing and cluttered.
Con No rich text editing
On Android you can receive messages with bold or italic text. However, you cannot bold or italicize text on Android, only on iOS devices or via the web interface.
Con No way to moderate groups
If a member of a group chat room becomes disruptive, or there's some other need, there's no way in Hangouts to remove somebody from an existing chatroom. Similarly, if somebody is no longer active, there's no way to remove that user from the chat room to make slot for new active members.
Con Buggy video calls
Whether video calls work on your device is pure lottery, and there's nothing you can do to fix it.
Con Poor indication of who's online
There is no way to find out if a person in your contact's list is online or not in the Android app.

Con Poorly manages data
Doesn't cache images well either received or sent. Often users will see the sent images re-downloading, same for conversations.
Con API support was removed
This change closed options for integrations and connections between similar apps.

Con Not available on Windows Phones
There is no Hangouts app for the Windows Phone.
Con Doesn't support sending voice notes or video recordings
Currently there is no built in function that supports creating and sending voice notes or video recordings.
Con No customer support
Free product, you're on your own.
Con Won't send SMS/MMS anymore
They removed support for sms/mms. The same goes with integration to hangouts.
Con No SMS/MMS backup
They don't have options to backup sms to cloud.
Con SMS messages sent from the web app can't be sent outside North America
SMS messages sent from the web app can only be sent to number in America and Canada - sending to foreign numbers won't work, although you can still receive texts from anywhere.
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
