Delete or Keep Previous Versions of PSE?

As John says, users sometimes keep older versions because they like to use features that no longer exist/have been changed in the newer version. The slideshow editor is a good example, but there are others.

Personally, I keep several versions on my computer so that I can help out folk with older versions on this forum.

One other point that is worth mentioning if you are concerned about saving space.  Your older catalogs will not work with the current version of Elements.  Hopefully, the old versions have been converted to the latest catalog.   However, if you use several different catalogs for each version, you will need to convert any that were not converted automatically with the updated version.  (Only the most recently used catalog will be updated when installing the latest version.)  Once you have converted the older catalogs, these can also be deleted.  Deleting the earlier programs will not delete the catalogs.

Some people like the old slideshow editor and keep older versions. But if that is not important too you you can safely uninstall your earlier versions.