You only need to pay the monthly fee if you sign up for the Retail version of Square. The Point of Sale app is free to use.
But to answer your question, the register check-out equipment is only a small part of the system. The majority of what Square provides is their databases and server-based systems. All of your inventory management, transaction records, financial reports, bookings, etc. are all actively and dynamically managed on Square's servers.
If you tried to put all of that data management on an iPad, you'd run out of memory fast, and you wouldn't have a backup of all your valuable data.
Basically, Square is a software service (although they provide hardware options as well). Software services generally cost a monthly fee to access and use.
