What differences are there between using the Server.app Time Machine server and simply sharing the attached backup HDD? I'll get a Mac mini next week and apart from other things, I'd like to use it as a backup server via Time Machine and an external HDD.

As I want to be able to access the HDD from remote anyway, I thought about simply sharing it and selecting it as a backup drive on my Macs. Then I found about about the Time Machine server included in OS X Server. What differences and advantages are there in using either one?
The  Time Machine functionality that OS X Server provides is to allow an remote OS X machine to use the server as a Time Machine backup disk. The server will create a sparsebundle for each remote machine it is backing up.

OS X Server includes a way of controlling and administrating services that can be implemented on normal OS X. In this case you can create a sparse bundle and share it in plain OSX (or as is done in many Network Attached Servers in Linux/BSD)

For your case as you are not using the server as a remote TM backup then OS X Server does not provide much here. With normal OSX you can share the external disk via AFP, samba but these are better not to expose across the internet for security reasons or better use ssh to remotely access the Mac.