In the Tesla app, the "connected" status of the Phone Key indicates that the phone's Bluetooth communications is in contact with the car. The phone needs to be within several feet for the connection to occur. When "disconnected", the phone cannot be used to walk up to the car and have the door automatically unlock; in other words, operate as a "key". That could also happen if you turn the phone's Bluetooth off (Airplane mode).
You CAN control all other remote operations of the car IF the car has cell service (AT&T). That's what provides the vehicle's location to the app. You can unlock the car, open the windows, turn on the climate control, etc. All of this is done through the cell connection. It is independent of the Bluetooth connection.