From a technical perspective there is no difference between the two. From an agency point of view the terminology is used to indicate the purpose of the environment. For example a development environment is where the developers themselves actually work. The code may or may not be completed and may or may not have been tested.
The staging server is where the code is deployed so that your clients can take a look at it Before it goes live. It must have the same configuration settings as the production environment and the software can be tested in live conditions.
StagingPro only distinguishes between the two during the setup process to keep inline with agency terminology but these names can be edited to make them more meaningful to you and your team.