That depends. If you have a strong cellular signal in your house and a large enough data plan to meet all of your needs, no you may not need to have home internet service.
If, on the other hand, you don't have a strong cellular signal in your house, have a small data plan or have devices that need to be connected to the internet all the time (e.g. alarm system, smart thermostat, etc), then home internet service is a good idea.
Verizon is always adverting cheaper rates on home internet service if you combine it with your Verizon Wireless service. I don't know the specifics but it might be worth calling and asking about it.