I have some reservations about the orbital mechanics of an asteroid being weeks away and approaching as slowly as it did whilst still being too close to be diverted. Even a relatively small force acting over a long period has a measurable effect and the Enterprise could apply a lot more than a small force. Also, its weapons have been described as being able to lay waste to a planets surface - it could certainly have carved off chunks small enough to be handled by the tractor beams.
Tell that to the folks who wrote "The Paradise Syndrome." There they had two months and weren't able to change the asteroid's course even a little. The best available fudge is that both asteroids were simply too massive. After all, a planet's surface is only a thin veneer, a fraction of its total mass.