Asie from the tides being gone (and the oceans probably levelling out), its been theorized that the Earth's wobble would also go out of balance. Not right away though, but during the millions of years after it, the Earth's wobble would become unstable.
That is, if the moon was completely gone. If it was like, blown up, the pieces would probably circle earth as a ring system and over time would form a new moon, just like it did during its creation.
Actually, the moon is really slowly moving away from us. After its formation it was so close it filled a good portion of the night sky, now billions of years later, it's a much smaller disc.
That won't happen for another few billion years though, so no life would actually experience this, since life on Earth will probably die out in about 500 million years-1 billion years when the sun will start to become a red giant.