tl;dr - you may need to exchange the VZW Note 4 for a T-Mobile variant to get the most out of your T-Mo plan. It may not even cost you a lot.
very long technical explanation if you want to try to make your current phone work...
All the different carriers use different frequencies even for the same technologies. Even if the VZW Note 4 is an "International" phone and supports GSM, that doesn't mean it necessarily supports the frequencies for DATA over the various GSM technologies (EDGE, HSPA, LTE).
I have an AT&T Note 4 I am using on MintSIM (T-Mobile). I have zero overlap with T-Mo's 3G frequencies (HSPA/HSPA+) but EDGE worked fine. I also had to noodle around in a service menu to enable LTE Band 4 so I could get 4G/LTE service. I won't have access to T-Mo services everywhere they offer them - only where they happen to offer LTE on Band 4 and EDGE on the few frequencies my phone and their service happen to overlap - fortunately where I spend all my time they do.
It could be that the Verizon variant simply doesn't support enough bands to work on T-Mobile except for basic GSM services like voice and text.
Turn off WiFi and go into "Settings" / "General" / "About Device" / "Status" [menus may vary slightly]
Do you have a "Mobile Network Type" and if so what is it?
Is your "Service State" reading "In Service"? Is your "Mobile network state" reading "Connected"?
Here is my status, connected to MintSIM from an AT&T model Note 4 with WiFi off.
