Depends on what info. SIM cards aren't used to store much these days, mostly relegated to your phone information and basic contact list. If you're still storing your contacts on the SIM card, which I strongly recommend you don't, then you'd have to move them to a phone or cloud first and then transfer to the new one.
As for phone info, the carrier needs to assign the new SIM Card address to your phone or it won't work.
That being said, if you MUST keep your SIM card, you can just cut the S4 one into a nanoSIM for the S7. Instructions on how to do it abound online or carriers can do it for you, too, most of the time.