The mainspring provides all the mechanical energy. The small spring is called the escapement, it spins one way, then back the other at a set frequency (typically 4, 6, or 8 times per second), letting the energy of the mainspring 'escape' gradually (instead of unwinding all at once, in a rush).
So the small spring is what makes it run accurately? I wonder how hard it is to make the small spring accurate without modern tools (1800s technology).