The Help section of ArcGIS Explorer Online describes the site like this: "ArcGIS Explorer Online is an online application that lets you explore and present maps within an efficient and well-structured environment." This site is different from ArcGIS.com in that ESRI considers ArcGIS.com a lighter version of ArcGIS Explorer Online. Also, ArcGIS Explorer Online requires the Microsoft plug-in Silverlight.
Note: Free ArcGIS Online accounts do not work with Explorer. It seems that Explorer only works with Enterprise or Corporate accounts.