The question is the same as what is the difference between buying something and paying for something.
One can buy a product. You can not buy a service, but you can pay for it. You enjoy a service while your subscription is in play, but at the end of your contract the service stops and you own nothing as a result. You can buy a car or pay for a taxi ride.
Some times it is a point of view. Sky own a number of TV channels, to them they are products. But when you pay to have access to a channel you are paying for a service. It seems that a thing can be both a product and a service at the same time. This is true. But what it is to you depends on your relationship to the thing. Conceptually if you could buy it and perpetually own it then to you it is a product.
The concept gets a bit more tricky in a case like buy a TV vs. rent a TV. In both cases the TV is a product. But in the first case you buy the TV and in the second case you pay to use the TV. In the second case you are both paying for a service and using a product.