Windows does not like to run on 2 interfaces at the same time. I am assuming you have the laptop plugged into the netgear and the printer is also plugged there.
open a dos window and issue the command ROUTE PRINT
You should see ip addresses for both your lan and the verizon. you should see routes for 0.0.0.0 in the table 2 times with different gateways and metrics. The metric is how it determines which to use.
You should be able to use the ROUTE ADD command to add your printer IP using a 255.255.255.255 mask and point it to the gateway for your lan. Since there is not a corresponding route for the verizon it should use this entry.
I assume if you turn the verizon off it allows you to surf via the internet and print and as soon as you turn it on it breaks the printer again?
Note you are getting very lucky since many times the metric for the ethernet is better (ie lower) than any other adapters. A lot of machines will ignore any other interface when you plug the ethernet cable in. You can change this metric in the interface settings.
It is quite a challenge to get a machine with multiple nic cards to function but mostly it is understanding the ROUTE command and how it selects routes from this table.
For 3 years my Pantech 190 modem has been plugged in to my Dell desk top and internet access for my Dell laptop has beem shared thru a Netgear router .
I purchased the new Verison 290/4G modem. It workrd great until I tried to print on my HP printer thru the network. No luck.
Verizon says that the 290 is not a dial up modem, but instead looks like a network adaptor and the desk top doesn't want to run two networks - the Netgear and the 290. In other words, the 290 will not work on a system that is connected to any other network.