Only if there are features on the new EVSE that you want, or if you want to carry the Nissan EVSE in the car for 'an emergency'. The Nissan EVSE is pretty basic but if it suits your needs, it will charge the car just fine. If you want wifi connectivity, amperage level settings, etc, etc you will need something fancier.
Also, I'm not sure if the Nissan EVSE cable is designed to be installed outdoors and repeated insertion cycles will wear out any normal 240V receptacle. If it was mounted inside, I'd just leave it plugged in continuously, if it's outside, you will probably want to provide some sort of protection from the weather (wet and damp).