This is a great question.
One of the biggest changes would be related to life on this planet.
Because of the moon's motion around the Earth we have tides. This narrow range of land which the tides affect is a very active area of the world in terms of the development of life.
In a world without a moon this section would not be very large and would not be changing twice a day. This influx and egress of salt water is hypothesized to be one of the main drivers for organisms to force adaption strategies and to evolve.

Also there is the effect of these tidal forces on the planet. Studies show that this cyclic stretching of the earth generates heat which drives tectonic plate motion. If we did not have a moon then our planet may have cooled significantly earlier and become dormant like Mars - whose two moons are too small to generate much of a tidal force on the planet. (see ref below)

And at night we could only see by starlight.... :-)

Here you can see the relative sizes of the moons in our solar system relative to the Earth.
http://i46.tinypic.com/wrgb5y.jpg

The length of the day, the tilt of the planet, the number of meteors impacting the surface are just a few of the things that would all be vastly different if the Moon did not exist. Not to mention all of the familiar cultural references that would no longer make sense.

Those theories really have no science or evidence to back them. The most major change to the Earth would be that without a moon the days would not continue to become shorter, as the moon slowly receding away from the Earth is slowing down it's rotation.