The "Miller" 1381 solar engine circuit problem
Right. Your problem is that you're using a motor which needs more current than the one in the article. In order to turn on the NPN hard (which will provide minimum voltage across the transistor and maximum voltage across the motor, you need to supply more current from the TC54 than it could provide. With inadequate base drive there was a fair amount of current through the motor, but not enough to get it turning. At the same time, the voltage across the transistor was high enough that the power dissipated in the transistor (voltage times current) became objectionable.

When you switched to a TIP120, you moved to a Darlington transistor, which has much greater gain. The greater gain allowed the detector output current to turn the transistor on much harder than the 2N3906. This allowed more voltage across the motor, and more current through the motor. 

I suggest that you measure the motor current when using the TIP120, and I expect you'll find significantly more than you got with the 2N 3906.

There is one drawback to using the TIP120. Since it's a Darlington, its saturation voltage will be about 0.8 volts more than a 3906. In other words, if you could drive the 2N3906 harder, you'd find the collector-emitter voltage to be about 0.2 volts, but the TIP120 won't go lower than about 1 volt. This means that, if you could drive the 2N3906 harder, you'd find it stays cooler and provides more current in the motor than the TIP120. Since you couldn't drive the 2N3906 hard enough, though, this potential superiority of the 3906 never became an issue.