When comparing Skype vs Matrix, the Slant community recommends Matrix for most people. In the question“What is the best chat software?” Matrix is ranked 1st while Skype is ranked 24th. 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 Easily connect with others due to how popular the service is
There's a good chance that whoever you're trying to reach has Skype already set up and familiar with it, avoiding the hassle of installing and getting to know new software.
Pro Screen sharing
You can show your whole desktop, a select part of the screen or a specific window over Skype.
Pro Cross-platform
There's software for Windows, Mac & Linux and apps are available on iPhone, Android, and Windows Phone.
Pro Group video for free
You can hold video calls with up to 10 people.
Pro Available on Linux
A stripped down, minimalist version of Skype is available for Linux.
Pro File sharing
Skype allows you to send any files (MP3s, images, text, etc.) by simply dragging and dropping then in the chat window making it simple to share things with colleagues.
Pro Cheap fixed phone number
Starting at $5/mo you can get a Skype number.
Pro Send SMS and make regular calls
With Skype Credit bought or a subscription, you can send SMS and make direct calls.
Pro Group chat with archive / records
Pro Powerful 1:1 chat and also group chat
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 Plagued by bugs on all platforms
Crashes and other bugs are prevalent on Android, Windows and other platforms.
Con Subpar video quality with more than 5 users
Skype has significant problems with multiple users and video quality. Skype recommends that a max of 5 users group chat as video quality is severely affected. Even webcams tested with other chat services showed lower quality when used with Skype.
Con Not a feasible alternative as a professional team chat
Missing key feature such as centralized mgmt, centralized billing, data retention policies, ...
Con Poor quality when screensharing
It is not possible to manually set resolution or bitrate, and Skype often defaults to a resolution so low that text is unreadable. It is also not possible to transfer control over mouse and keyboard.
Con Slow filesharing speed

Con Owned by Microsoft
Con Obnoxiously large ads
Ads take up over 25% of the usable space.
Con Poor Linux support
The version available for linux users is quite outdated, and does not support some of the newer functionality, such as the free group video calling and screen sharing.
Con Requires both ends to be online (if neither end is using a cloud-enabled device)
Messages are sent and received only when both ends are online, if neither end is using a cloud-enabled device. In this case, if you are online 9-17 and your recipient is online 18-3, messages will never be delivered.
Con MIcrosoft (Skype's owner) actively assists eavesdroppers
On the other hand, if you lose your copy of one of your conversations, you may be able to get a copy of it from your government with a little bit of luck and a public records request.
Con Keys cannot be checked automatically
You cannot automatically check keys of your recipients. Only manually.
