Sprint, Boost, Freedompop, Net10, are Verizon resellers. Here is a list of all. You will see two types of network: GSM(AT&T) and CDMA (Verizon). The network technology is not the same. Therefore an AT&T phone does not/cannot work on the Verizon network. Also unless you are a LTE custormer, you may well be on Verizon's slower 2G or 3G network.
https://en.wikipedia.org/wiki/List_o...work_operators
Isolated areas are very different. I've lived in them for many years. I currently live 50 miles from any town of size yet we have cell service from US Cellular, Sprint, Verizon and AT&T. All work here with few if any problems. However, when I frequently traveled across these United States I became painfully aware of areas that only had one cellular service.