I was thinking about the fact that all the largest Solar System moons are tidally locked to its primary and this question arose.
What would be the practical consequences (on earth) if the Moon was not tidally locked?
Honestly, I think the consequences would be pretty small, except we'd see the dark side of the moon from time to time. On the moon, the consequences would be bigger.
All tidally locked means is that the moon's rotation matches the moon's orbit, so that the same side of the moon always faces the earth. If the moon wasn't tidally locked, it would spin from our point of view. The moon spinning wouldn't affect the earth hardly at all - at least, in no way I can see.
The reason most moons are tidally locked to their planets is because the planets gravitation on their moons is quite large. Strong gravitation, or, strong tidal effects is perhaps more correct, slows down an orbiting objects rotation, so tidal locking of moons is common. Tidal locking of planets - less so. Both Pluto and it's moon Charon are tidally locked to each other cause they're pretty close to each other. Mercury is also, nearly tidally locked to our sun.
Now, Phuc's answer
By being tidally locked, the Moon has been extending the Earth day by slowing down the Earth's spin, to about ~6 hours from an 18 hour day to a 24 hour day.
As HDE pointed out, this isn't so. It's the earth's rotation being ahead of the moon's orbit that's caused the earth to slow down. The moon's tidal effect on the earth plays a role in that, but the Moon being tidally locked to the earth is irrelevant.
Also, the earth's spin was much much faster than an 18 hour day when the moon was young. By this article, a day on earth was only a few hours long. http://sservi.nasa.gov/articles/nasa-scientist-jen-heldmann-describes-how-the-earths-moon-was-formed/
The earth, 4 billion years ago was spinning unusually fast for an object in our solar system. It might help to consider what makes planets spin. When they form, it's conservation of angular momentum, but according to the giant impact hypothesis, the earth was hit, not dead center but at an angle. The giant impact that formed the moon also set the earth spinning very fast. The moon was also very close when it formed - maybe just twice the Roche limit, so, that close, that the moon slowed the earth's spin and the (at the time) much larger tidal effects pulled on the moon, causing it to move farther away.
If the Moon was not tidally locked, it would mean that the Moon is in the wrong orbital distance from Earth. The Moon is tidally locked because it is close to the Earth. If the moon were closer, it would approach the Earth's roche limit, be torn apart, and its debris would become a ring for ~100-200 million years. If it were too far, it would continue to drift outward until it was free of Earths' gravity.
By being tidally locked, the Moon has been extending the Earth day by slowing down the Earth's spin, to about ~6 hours from an 18 hour day to a 24 hour day. This process will continue into the future until the Earth is also tidally locked with the Moon. The Moon, while in this state, has also stabilized the Earth's seasons by stabilizing the Earths' orbital axis.
Lastly, a possible indirect consequence would be upon human history. Mankind would have probably abandoned the celestial sphere model in favor of the spinning Earth model much sooner had they seen proof that a celestial body could spin and rotate, much like when Galileo saw that moons were orbiting Jupiter.