Q: What are the advantages/differences of the modern Skype app over the desktop version?
A: Generally desktop and modern apps are slightly different approaches for creating and distributing programs. Microsoft is trying to apply Apple's and Google's with it's Google play store ideology and to distribute software through centralized Window's app store. Usually it is easier for user to install and to upgrade application through centralized repository. On the other hand this puts some limitations on applications itself, e.g. MP3 Skype recorder can't be modern app as it requires access to disk and low level access to audio.
Skype modern app offers slightly different interface and lacks some features, e.g. you can't open a call/conversation in separate window. You can't open two instances of Skype app to use with different Skype accounts etc.