Answer: In an FET, the gate is insulated from both the source and the drain by a very large resistance. Hence, there is no current flow into the gate. Therefore, it must be a voltage-controlled device. Specifically, the channel resistance is a function of the gate to source voltage.  

In a BJT, the base is connected to the emitter via an internal resistance. The resistance is in the low ohms. Therefore, there is a means for current flow. Hence, the BJT is a current-controlled device. Specifically, the current through the device is a function of the current into the base.