Circuit with a tip120 works with a Led but, don't with a vibrator motor
The TIP120 is possibly the worst transistor you could choose for this job. It is a Darlington Pair transistor, and those have a (relatively) massive forward drop of between 2 and 4 volts. It works with the LED because an LED needs a tiny amount of current to operate compared to the motor and has a fixed forward voltage of its own. The motor doesn't, and is being starved of power by that huge voltage drop (it's getting maybe half the voltage it should be getting).

Throw away that TIP120 and replace it with a decent N-channel MOSFET - one with a "threshold voltage" (VGS) of 2.5V or less ideally.