Virtual Desktop Infrastructure (VDI) is an individual virtual machine running a desktop Operating System. Although there can be multiple virtual machines per a given underlying physical server, but, only one user at a time on a given virtual machine i.e each user can be assigned their own virtual machine.
A Session-Based Desktop is an individual session running inside a server Operating System on either a virtual machine or a physical server. There can be multiple user sessions per virtual/physical machine i.e. each virtual/physical machine is shared by multiple users.
Danny Allan (Desktone):
VDI may mean hosted desktops. But, contrary to conventional wisdom, it doesn't mean those virtual desktops can be effectively hosted in the cloud. VDI and cloud-hosted desktops are very different animals. For instance, VDI requires significant upfront CAPEX, is complex to implement, results in vendor-lock in, and doesn't scale easily.
Conversely, a true cloud-hosted desktop infrastructure is technology agnostic, designed for multi-tenancy and service grid architecture - so enterprises get up and running quickly with no CAPEX, and can easily adapt the environment to their changing needs.