Technical explanation of why *explicitly* 13.1?
Cause it is in fact the whole 13.X driver branch that is broken, and I am just mindboggled as to why 13.1 needs to explicitly be used.

Also, AMD have since a while now, released their own tool for cleaning up the drivers. So a third party tool is no longer necessary.
oclHashcat is not open source and thus ships pre-compiled kernels. The kernels are compiled for 13.1 and will not work with other versions. Although 13.4 seems to be close enough to 13.1 that a lot of users are reporting that 13.4 works fine. Just don't file any bug reports if you use 13.4 since it's unsupported with the release version.

Not a Windows user, but if you search the forums a lot of people report that AMD's driver cleaner tool doesn't work properly, hence the recommendation for Driver Fusion.