Brightness published for this comet in the article is in "magnitude" and it's a logarithmic value of total brightness as used in astronomy. Not brightness per unit area.
So if the comet is much smaller than the moon, is it reasonable to assume that it wouldn't reflect enough light to light the earth's surface like a full moon?
If the comet is much smaller than the full moon, then its surface will be brighter than the surface of the moon to still be of higher magnitude in brightness than moon.
Magnitude is total visible luminosity of the body. Think of it as a logarithmic value of lumina. Total amount of visible light from that whole body that is visible to an observer on earth.
Watts is not a useful measure of visibility for humans, I would hope they're using lumens. Unless it's safe to assume that it reflects very similar amounts of visible light to the moon, but I don't know if that's the case.
In most contexts. But it's used frequently in "infrared light" too. Watts include radiation across the entire EM spectrum, from radio up to gamma rays, while lumens are normalized to human perception.
You basically take the number of watts at each wavelength and multiply it by a factor that represents the eye's sensitivity at that wavelength. So 1 watt of IR is zero lumens, and 1 watt of blue light is fewer than 1 watt of yellow.
Take a look at the spectral distribution of daylight [1]. You'll notice the area under the curve in IR is pretty similar to the amount of visible light.
If the IR reflectivity of a comet were high and the visible were low (relative to the moon), knowing that they were reflecting about the same wattage wouldn't tell you much about relative brightness.
Comets don't reflect light, they generate it moving against solar wind and radiation. After those things are left behind it becomes a just a big ol' flying rock again.
To expand on this a little - the solar wind vapourises part of a comet, forming an atmosphere of sorts called a coma (and the characteristic tail) and this atmosphere glows both by reflecting sunlight and by ionisation. So yes, the cometary nucleus hardly reflects any light, but some light is reflected by the coma and the tail.
(see en.wikipedia.org/wiki/Comet#Coma_and_tail, and just above that for details of light reflection from a cometary nucleus)
The (total) magnitude is just a convenient scale to measure of how much light that is radiated/reflected on an object. If the light was radiated equally in all directions, the light (all photons) would after a time have travelled equally far, defining a sphere. Since the photons travel further away every second, the surface area on the sphere increases as r^2 (r=distance to object), but the number of photons is the same.
That means that the light density, or the number of photons that hit our eyes/cameras, will decrease with the same factor, r^2. That's why they created the apparent magnitude. The (total) magnitude is a measure of how much light that leaves the object, while the apparent magnitude tells us how much light will hit the earth, or equivalently: how bright will this object appear on the earth.
In other words: If the comet is 20% the size of the moon and the same brightness, is it throwing off 20% as much light due to the lower surface area?