Simple answer : there would be twice as many fake moon landing theories.
It would be interesting, one useful thing about the moon being that it stops earth wobbling too much on its axis, hence preventing drastic climate change. Though i've only really heard discussion about if earth had no moon, basically sayying that it needs one for axial stability. Though half of the planets have more than one moon in this solar system. Are they that ustable.
The main thing is, it depends on the moon. A small asteroid like Mars's moons wouldn't be that important. A large moon like Titan? Its almost a as big as earth, a binary system could acutally form. And that would fudge up a number of things, including the magnetic field which stops us getting 'toasted' by solar flares.