When comparing Google Hangouts vs Matrix, the Slant community recommends Matrix for most people. In the question“What is the best chat software?” Matrix is ranked 1st while Google Hangouts is ranked 15th. The most important reason people chose Matrix is:
Matrix is an open standard, defining simple HTTP APIs so that devs can easily write their own clients, bots, bridges or servers. You're not locked into a specific set of implementations.
Specs
Ranked in these QuestionsQuestion Ranking
Pros
Pro Mobile support
Hangouts offers both an Android and iOS app so users can participate from nearly any mobile device in addition to his Desktop.
Pro Screen sharing
Any user can share his screen (or only a specific desktop or window) while still continuing an audio conversation.
Pro Free group video support
Hangouts works with up to 10 participants using video out of the box or up to 15 if you are a business, government or school. It determines who is the primary speaker at any given time, and the camera view automatically switches to that user.
Pro Useful for businesses
Hangouts can scale to become a videoconferencing solution for an entire company, since the software can run on nearly all platforms. In fact, Google itself uses Hangouts as its meeting room video conference software.
Pro Easy to start using
Hangouts is integrated with Gmail & Google+, does not require you to download desktop software, and all chats can take place directly in your browser, significantly reducing the barrier to entry for a new user.
Pro Completely free
Google offers all features of Hangouts completely free. There are business, education and government solutions that increase the participant count to for video calls to 15 and offers support.
Pro Cross platform
Hangouts work on Android, iOS, Chrome OS, web and as a Chrome extension.
Pro Live broadcasting and instant recording
With Google+ “Hangouts on Air”, you can broadcast any hangout to an unlimited audience via your Goolge+ Profile or simply a generic YouTube link. Moreover, any Hangout on Air (public or private) is automatically saved as YouTube video that you can keep for private reference or use in any way that you would normally share a YouTube video. This allows you to record conversations without additional obtrusive software.
Pro Google Voice integration
Allows for a fixed phone number, receiving and making calls. Calling US or Canada is free.
Pro Integrates with Google Maps for sharing locations
When chatting with people while trying to plan a meet up, the share map button can be extremely useful.
Pro Android app does both sms and hangouts messaging
Pro Built on an open standard
Matrix is an open standard, defining simple HTTP APIs so that devs can easily write their own clients, bots, bridges or servers. You're not locked into a specific set of implementations.
Pro Bridges other networks into a single decentralised network
Matrix has bridges to IRC (freenode, moznet, oftc, snoonet etc), Slack, Gitter, Rocket.Chat, XMPP, SMS, SIP and others. The point is to 'matrix' all the different networks out there into one single decentralised network.
Pro Has an easy to use client called Riot
Riot.im is the easiest way to use Matrix, with great clients for Web, iOS and Android (and Fdroid).
Pro Does not require a centralized server to establish a connection between two users
Matrix is decentralized, there's no one central point that the information goes through and so no once central point of failure or control.
Pro Matrix prioritizes direct messaging with people the same as Slack-style groups
Matrix aims to "provide an analogous ecosystem to email - one where you can communicate with pretty much anyone, without caring what app or server they are using" using a neutral identity system.
Pro Supports different kinds of communication
Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling (voice and video) and Internet of Things communication.
Pro Has an app store for 3rd party integrations & bots
Riot.im includes an app store with integrations for Github, JIRA, Jenkins, Giphy etc - and anyone can add more via Matrix.
Pro Offers choice of clients
Which can be found here.
Pro Maintains full conversation history
Pro Has an active community behind it
Pro Can be integrated with existing communication services
Matrix is designed to support Instant Messaging, VoIP/WebRTC signalling and Internet of Things communication and allows cross-communication between those services. Meaning one person could be using IRC and another Slack for IMs, or one person could be using Skype and another Google Hangouts for videoconferencing.
Pro End-to-end encryption
Matrix features end-to-end encrypted chats which are syncronized accross all your devices.
E2E is implemented in the matrix-js-sdk library and Riot.im client.
Cons
Con Poor video call quality
Only goes up to 720p30 + low bitrate.
Con Owned by Google
Con Lacking clients for older versions
Con High hardware usage
Con Not recommended for long sessions

Con Won't work on Windows 8-style UI
As Hangouts uses a browser plugin on the desktop and it doesn't work with Windows 8-style UI, it has to be used in desktop mode.
Con Keys cannot be checked automatically
You cannot automatically check keys of your recipients. Only manually.
