A wire connected to a bulb does not glow, whereas the filament of the bulb glows when the same current is passed through them. Why ?

Filament of the bulb and the supply lines are connected in series. So the same current flows through them. Since the resistance of the connecting wires is negligibly small as compared to the resistance of the filament and heat produced is directly proportional to the resistance, therefore, the heat produced in the filament is more. Hence, the bulb glows, but the connecting wires remain practically unheated.