Notionally, both 고객님 and 손님 have the same meaning of "Customer" But in my personal opinion, 고객님 sounds even more formal since 고객님 is mostly used in certain places such as department stores or customer counseling offices while 손님 is more widely used and 손님 also can mean "Guest" or "visitor" ex) 어제 저녁에 우리집에 손님이 오셨다. We had a guest in our home in the last evening.

I have a same opinion with Ahtti. In addition, to me, 고객님 sounds more polite and is used when a seller is selling a service, which needs signing on a contract, to their customer.

There is no difference. You can use both of them, when you want to say.