Switching to some apps in a different virtual display doesn't work
The setting System Preferences > Mission Control > "When switching to an application, switch to a Space with open windows for the application" must be enabled for OS X to automatically switch your desktop/Space when changing to an open program.

Otherwise, it will stay in the current space and one of a couple of things will happen, depending on the way the application is programmed:


The menu bar will change to that program's, but nothing else on the screen will change, or
A new window for that program will be opened (think a text editor opening a new, blank document), along with the menu bar changing