The Samsung S3 comes in different variations depending on the Carrier. Some are only GSM and 4g (AT&T,T-Moble), some just CDMA and 4g (Sprint). Verizon S3 has CDMA, GSM and 4g. Even the 4g radios are different. The international version has a different processor. When you purchase an S3, you need to be aware of the model number to know what is inside the phone. You must be careful to root only with the correct version firmware.
One great advantage of the Verizon S3 is that GSM is unlocked and with the latest Verizon June 20 update you can update the APN settings when you put in GSM sim. This works not only overseas but in the US you can even use AT&T and T-Mobil sims. You just cannot get 4g on AT&T or T-Mobile.