The online free version of Photoshop Express really should go by a different name. Even with the "express" moniker, it does not do the desktop software justice. It does serve a purpose, but it is not for extensive use or a rigorous workflow full of hundreds of images.
I was pleasantly surprised by the speed and capabilities of the online suite. It is very nice to use for a quick edit if for example you were away from your computer or on a lower powered machine that does not handle the software well. It does have limitations. It is closer in line with the Photoshop Express for smartphones(iPhone, Android), then it is in line with the desktop Photoshop Express software.
To directly answer your question, the advantages of the desktop Photoshop Express software are currently almost too broad to answer. It would read more like a list of what Photoshop Express does, and not what the differences of the two pieces of software are. The online version does very simple image manipulation that all photo editing software has done for years, and it does it in a online environment that will limit the speed and functionality of a photo workflow.