iOS9's new Split View and Slide Over let you use Skype while having open another app alongside it. This makes browsing and sharing images and other content easier during chats and video calls.
The improved notifications feature allows to reply to incoming messages directly from the lock screen. To use this optional feature it has to be activated in Settings > Notifications > Skype > Allow Notifications. You can just opt-in when you like the idea or just stay with the old behaviour.
A better integration into Spotlight makes it now possible to start Skype conversations directly from search results. Existing Skype contacts will show up automatically, a Skype chat can be initiated tapping on the name of your contact.