Some points of all 3
Discord:
Multiple Servers
Multiple Channels
Ranks and permissions (and server groups)
Friends
Group Chats
Calls (group and single)
Free (Even to make a server)
Somewhat buggy, not as much as Skype
Stream friendly
Overlay
Emotes
Status (Away, Invisible, Do Not Disturb,Online)
Skype:
Group chats
Friends/Contacts
Calls (group and single)
Status updates
Most buggy of the 3
Emotes
Status (Away, Invisible, Do Not Disturb,Online)
Screen Sharing
Teamspeak 3
Servers (Not free)
Channels (Plus descriptions and images in the channels)
Easy use of plugins
Away status (custom ones too)
Multiple servers
Contacts (Eh, not really the same way as the other two)
Overlay
Few emotes
In my experience, Skype is by far the most unreliable, Teamspeak is ok, but Discord is by far the best. It takes from the best of both and puts them together in a free platform. Skype and Teamspeak are both Technically free, but have another fees/payments associated with them (optional). Far easier to just create a Discord server.