It drops Flash in favor of Encrypted Media Extensions (EME) for music playback, which is supported natively by most modern browsers. Check to make sure your Flash version is up-to-date. So in order to play songs in the application, make sure you have an up-to-date Adobe Flash plugin.
The new Web Player is in line with the overall Spotify look and feel and is built on HTML5 standards. Spotify Web still uses Flash over the HTML5 player. PS how can you STILL not sort the playlist display alphabetically, or for that matter sort a playlist (i.e. The simpler and faster web player outperformed the old web player in all key metrics. PS: I also assumed this player is developed in HTML5, and not implemented using the security vulnerable, memory intensive Adobe Flash. One defines an MP3 track and the other defines the OGG format.