OK, that seriously makes no sense. 
You’re saying that if your phone is on the Wi-Fi network, but it still has Internet access, you could open a browser and go to the news or whatever, but when you open the SmartThings mobile app and go to My home, and look at the things list, and find the GE wall switch, and tap the icon that that wall switch does not change state.
But that if your phone is on your cellular network and you do exactly the same steps, the wall switch does change state.
I can’t even imagine from an engineering standpoint how that could happen. I believe you, and SmartThings has surprised me before, but that one is just really weird.
Fortunately these are officially compatible devices, so support should be able to help. I would just get in touch with them at Support@smartthings.com and see what they can figure out.
If you want to look into the logs on your own, you can, but I’m not sure you’ll see anything that will help diagnose it