The car can either connect to wifi or to the cell phone network, never both at the same time. Sounds to me like the car is connecting to your wifi, but the signal is not strong enough to function. However, since it is connected, the car does not even try to use the cell phone network.
Move the car away from the house and see if it connects to the cell network once it is completely out of range of your home wifi.
You will have to work with your wifi to make the signal stronger or get the car closer. Realize that the wifi antenna is in the passenger side mirror. Sometimes reversing the direction of the car or orienting it so that the passenger side is facing the house will help. Often it requires additional equipment to make your wifi signal stronger, though.