How could Piccolo blow up the moon, while Goku barely managed to leave a small crater? In Dragonball Z, when Gohan changed into an Ozoro, Piccolo found an appropriate solution, destroy the moon. Easy-peasy.

The real question is, when Goku used his Genki Dama (on two occasions, by the way), it never resulted in anything more than a relatively small crater. With all the power stored inside that sphere (easy above a billion, and over 9000), shouldn't it have caused a bit more damage? Piccolo managed to destroy the moon, easily, with power level of less than 1000.
Piccolo isn't the only character to blow up the moon in Dragon Ball. Master Roshi (disguised as Jackie Chun) destroys the moon at the end of the 21st World Martial Arts Tournament using the MAX Power Kamehameha, which takes considerably more effort than for Piccolo. In the meantime, Kami recreated the moon, so this isn't a plot hole.

However, it's a good question as to why the moon is so easy to destroy, while the earth is so much harder. Part of this can be explained since the moon is much less massive, but even then the numbers don't quite add up if we assume the Earth and Moon are the same as ours. Let's assume that physics in the DB universe is at least somewhat similar to our own, modulo spirit attacks and whatever else. Of course, this assumption might be wrong, but if you start changing how physics works, there's no reason why the moon can't be filled with TNT so that it explodes whenever anyone so much as looks at it.

I've made some approximations in the following calculations. I calculated the gravitational binding energy of the moon to be 1.2 * 10^29 J. That's the minimum amount of energy to blow up the moon so that it won't reform by its own gravity, so we can assume that the characters were using at least that much energy. That amount would be enough to knock 3.4 * 10^20 kg of matter off of the surface of the earth, which is roughly 0.006% of the mass of the earth. That doesn't seem like a lot, but if we assume a uniform density of the earth, that's about 1.6 * 10^7 cubic miles of matter, which corresponds to a cube where each side has length 250 miles. That's a pretty huge crater. So just saying that the moon is easier to blow up than the earth isn't going to cut it.

There are 3 ways to resolve this. First, the earth and moon in the DB universe could be very different from what they are here. Specifically, the moon could be larger and less massive, and the earth smaller and more massive than in real life. However, it takes quite a bit of stretching to get believable values, and that's ignoring the vast power difference between DBZ Goku and DB Roshi. If you push things that far, I wouldn't expect it to be consistent with everything else in the series. The second possibility is that the moon is somehow more vulnerable than the Earth to attacks. This is roughly what Alenanno's answer says. It's entirely possible, but as far as I can tell there's little indication of this in the series apart from the Genki Dama, so I don't know why other attacks directed at planets would not destroy them.

This leaves the third possibility, which is my personal choice: the producers weren't really thinking about it when they did this, and were more concerned with making things dramatic than making them consistent. As such, I'd call this a plot hole, but it's a pretty minor one in the grand scheme of things.