The spin is believed to have been about four times faster early in Earth history, such that days were only six hours long. Eventually, if the solar system lasts long enough, Earth’s rotation will become tidally locked to the moon, so that the length of one day will equal that of one lunar month.
Without the moon, Earth would spin faster, the day would be shorter, and the Coriolis force (which causes moving objects to be deflected to the right in the Northern Hemisphere and to the left in the Southern Hemisphere, due to Earth’s spin) would be much stronger. This would lead to air circulations in the atmosphere that might look more like Jupiter than what we see on Earth today – multiple east-west jet streams and likely more than one big gyre in the oceans. For example, the Gulf Stream would leave our coast far south of Cape Hatteras.
Those circulations might be less effective at moving heat from the tropics to high latitudes, leading to warmer tropics and colder polar regions.
Second, the moon is responsible for most – but not all – of the tides through its gravitational attraction of Earth’s oceans. To take an example from a tide table for Morehead City, it appears that the typical lunar tidal range, high to low, is about 2.5 feet, while the solar contribution is about 1 foot.
I just did a quick calculation and found that the tidal force on the Earth from the moon is 2.2 times stronger than that from the sun.
Without the moon, tides would be weaker, affecting the tidal ecosystems for which tidal mixing and flow are important sources of energy. But this seems more like a quantitative effect, not the huge qualitative effect we’d get from a much faster spin.
More subtle effects would be on organisms that depend on moonlight and that synchronize their nocturnal activities to the phases of the moon.
Obviously, with no moon, moths would not have evolved to use the moon for navigation – which might mean they wouldn’t be drawn to light.