The entire online (cloud computing in general) is moving towards a business model which requires no capital investment (i.e. you don't have to buy a physical machine, employ an IT staff, and purchase/install/maintain ArcGIS Server) to an ongoing operational expense (a monthly payment). The idea is also that the operational expenses will be reasonable due to economy of scale.
Of course, you can also license ArcGIS Server in Amazon Web Services and either use that by itself or in conjunction with ArcGIS Online. Server is a good product but it requires significant time and energy to maintain. I know that from experience but you can also see this by looking at the various problems posted in the server forums.
Online also offers a set of functionality with regards to sharing information that is not offered by ArcGIS Server but rather ArcGIS Portal. Other than that, ArcGIS Online is less mature that server and much less functional (e.g. no support for images other than as tile packages). Like server, I believe that ArcGIS Online will rapidly change in terms of both functionality and licensing as ESRI hones in on the right model.
This is good stuff!
