When should I use one versus the other? I want to cache a certain object on startup and reuse around the application. Which sounds like the better solution (ViewData or Session)?
ViewData is a per-request object used to send information from the controller to the view.
Each action invocation gets its own ViewData; the ViewData doesn't last beyond the view.
Session State is a per-user storage container which allows you to store data for a specific user session (identified by a cookie)
If you want to share a global object, you should probably make it a singleton (in a static property) or put it in Application state.
Make sure that it's thread-safe. (Or use a [ThreadStatic] field carefully)