What do HR companies do differently compared to an S1 shop?

You could be doing anything from learning about casualty operations, personnel accountability, or postal operations. Day to day life is different for every company. Take this time to do some college and/or correspondence courses.

Different companies do different things.
Some are postal units.
Some do theater support for a Corps. Stuff like leave/ E Leave, administrative clearing for units and individuals.
Some work for garrison support units and do things like in/out processing on post.

HR may vary according to the type of Organization and the level of service or product released or sold. HR is driven to support the Personnel files, the payroll, the interviews of new employees the offer packages for new employees the exit interviews for departure employees, arrange background checks for potential employees and many details from offering personnel growth packages, detailed vacation and holiday calendar arrangements, promotion support of internal customers to different departments or a lateral move to other potential positions as they come up available etc. there is so much that is also including the organization work statements and objectives and also provide support of the rules and regulations applicable to every employee curriculum or role and responsibilities appropriately available and defined.