When comparing Skype 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 Skype is ranked 10th. 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 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 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 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 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
