No moon = no werewolves. And then Bela and Jacob could...
Also, the moon landing would really be thought a hoax by later generations. "What moon?" "Who's Tupac?"
But something the size of this current asteroid wouldn't be big enough to knock the moon out of its orbit. Anything that could do that would probably break up the moon or else knock loose so much moon material that the earth would be bombarded by all that, which would be enough of problem.
It's not certain what would happen if, say, the moon just disappeared. Tides would be different and that would affect life in the oceans. Also, the earth's orbit might become unstable.
The moon is actually moving away from the earth on its own at about 1.5 inches each year. Centrifugal force, baby. Soon (2012) it'll move far enough away from earth's gravitational pull that it will leave orbit (and go to the moon).
Dogs then will have to howl at ...streetlamps, or clouds that remind them of the moon.