Can anyone actually answer the OP question (or at leas my reading of it)?
While everyone concentrates on the spec differences (i.e., extra drive and different software), what I (and i think OP) want to know is "why server?" When given the choice, why would I go for a server version of anything? And please do not tell me "because it has an extra drive, etc."
I fully admit my ignorance of the subject, but here is one possible angle: is it because I could host my own website that way? Again, my apologies if that sounds stupid, but just wanted to indicate the type of answer I am looking for.
Being a Computer engineer I would be glad to fill pages and pages of reason, but not knowing what you use or need, it's hard to tell -why/how- you should use it.
In general it's a low wattage Mac which could be used 24/7 unattended and reached via Screen sharing/file server when you need it.
Mail server, directory server, chat server, web server, file server, multimedia server, Final Cut Pro X cluster, you name it.
Any SoHo environment could be managed with a Mac mini, especially if your clients are Mac machines.
Mac mini and Mac mini server are essentially the same, where the latter has OS X Server pre-installed (which you could install an a mini anyway) and the second hdd. That's it.