Current will only start to flow to the base of the bipolar junction transistor once the base - emitter voltage gets above + 0.7 V or so. Thus only signals above 1.4 V peak to peak will get amplified and switch on the LED. While LEDs will light up practically instantly, an incandescent bulb takes time to warm up before emitting light, so the transistor has to conduct for a much greater portion of time before light is produced.
Another problem with your circuit is that there is no path for charge which has flowed to the base of the transistor to replenish, as the transistor won't allow reverse current flow through the base at low voltages. The audio input is most likely capacitively coupled at the source, so this DC bias current can't come from the audio source either. The end result is that the circuit will build up a negative DC voltage at the audio line, which makes it progressively harder for positive peaks in the audio signal to overcome the combination of the base-emitter voltage drop and the generated input DC offset voltage. Eventually just leakage currents trough the capacitor and transistor will be replenishing the charge at the base, which is apparently just enough to illuminate a LED.
The simplest way to fix this would be to add a 1uF or greater capacitor between the audio input and the base resistor, and a schottky diode between the ground (anode) and the base resistor - capacitor node (cathode). This way the capacitor will charge trough the diode during negative peaks in the audio, and discharge trough the transistor base during positive peaks. A normal diode will work too, but schottky diodes have a lower voltage drop (e.g. 0.2 V instead of 0.7 V), allowing for the circuit to be more sensitive.
If this isn't sensitive enough, search for "audio peak detector" on the web.