With onprem software, you put a copy of the software on your server or on your laptop, and every time you run the software you load a copy into the computer memory. Both those acts are forms of copying (and so are infringements of copyright) which means that you need a software licence for the use to be legal. That’s not the case for SaaS software which sits on the supplier’s servers and never on your machines. Your use of SaaS doesn’t involve you in making copies of the software and so the questions of copyright infringement, and needing a licence, do not arise.
But the important difference is the risk profile. Because SaaS software doesn’t sit on your machines, you are not independent of the supplier. If the supplier goes bust, or decides to switch you off, you are – in practical terms – up a creek without a paddle.
That’s a much riskier place to be than onprem. With onprem, the supplier can’t switch you off and, even if it goes bust, you will be able to carry on running the software without support for a decent period.
The position is the same with your data. With onprem software, your data resides on your servers and, if the supplier goes bust, you still have your data. With a SaaS supplier, on the other hand, the data resides on the supplier’s servers. If the supplier goes bust, or decides to cut you off, you are left without your data.
There are lots of advantages to SaaS compared to onprem, but the risk profile is not one of them.