I think other answers give you a pretty good idea of what the differences would be between a moonless Earth and our own. But there are a few points that I think might be highlighted, that seem to be missing.
As others mentioned, the Moon has no deal with Earth's seasons, aside from estabilising its axis. But the impact that created the Moon - if this is what happened - or even the capture of a small-size planet, if we believe otherwise - could have had whatever consequences a fiction writer wants. So you could position the South Pole of your fictional Earth in Africa - or whatever continent it would result in its place from the Moon not being created/captured - and explain that away by stating that what put Antarctica in the South Pole was the creation/capture of the Moon. The same goes for the length of the day: while the existence of the Moon slows Earth's rotation very slowly, the acquision of the Moon might have brutally reduced the rotation speed in one coup - or, on the contrary, accelerated a tide-locked Earth.
(Note that a moonless Earth would be more different from actual Earth if the Moon was indeed created by an impact; it would be unlikely that in that case the outline of continents or oceans would be remotely similar. If you don't want to rebuild our geography too much, it would be probably better to postulate that scientists are wrong, and the Moon was a neighbouring dwarf planet that got gravitationally captured into Earth's orbit.)
More importantly, though, the tidal effects of Moon affect all of Earth's liquid parts - most obviously the oceans, but also whatever liquid or semi-liquid parts of its mantle and core. This creates heath of itself - imagine the amount of attriction it generates - and also tends to stir the inner parts of our planet, making the decantation of heavier elements towards the core slower. In other words, a moonless Earth would quite probably have a bit less internal heath, resulting in reduced vulcanism, perhaps a deeper crust, even to the point that tectonic plates would weld into each others, and quite probably would have a crust less rich in iron and heavier than iron elements. And really heavy, radioactive elements would be even rarer, which would result in an even cooler interior, as part of the Earth's internal heath comes from radioactive decay inside it.
Nights would be darker. Would we have to adapt to this, too? And would other animals become more - or less - dangerous from the total absence of light at night, without a cycle of darker/lighter nights?
It would be interesting to see what impact the absence of the Moon would have upon human knowledge, supposing that your moonless Earth is inhabited by humans (but would it be, or, as Asimov suggests, reduced radioctivity would have slowed evolution too much?) Would realising Newton's laws be made more difficult without a close example of an orbit? Would time recording be too difficult for early men without a relatively easy-to-count 29 day cycle? Would we be able to realise the regularity of years/seasons without months and weeks to mediate/facilitate the calculus?
... and, of course, there's the pressing question of whether would there be such a thing as "romantic love" in a Moonless Earth.
