The phone worked fine with the fried card. All it didn't have was 4G. The card itself has the 4G logo on it.
The interchangeable sim card with your phone number may do that on other networks but not Verizon. I bought a T-Mobile phone on eBay so I could send my phonebook to my car (back in 2005) and used my Dad's sim card. Later I got an unregistered sim card (so I didn't need my Dad's card) and it cost $20.
When I went from the Bionic to the S4 the contacts came over via the cloud. They didn't just pull the card out of the Bionic and move it over.
I see where the problem is. The S4 works on several carriers and I first presumed that the OP was on Verizon. So yes, if it isn't Verizon you can put in the sim card you have and it should work. If it's Verizon, the sim card is just for 4G.
I'm on Verizon and speaking for phones on Verizon. Sounds good that it works on your network. I always get a new SIM card when I change phones due to it not costing anything. As a matter of fact, Verizon tech support usually recommend a SIM card change if a person has problems with their phone.