The moon is made up of a substance that emits gamma rays. It bathes the planet in low levels of radiation, and has done so since humanity's inception. It is the same size that it is in real life, and the same distance from earth. The atmosphere is also similar to ours. How would this form of radiation benefit humanity and the rest of the planet?
I freely admit I'm reaching here:
It would make the moon unsuitable for an alien invasion fleet to set up a command base there. Their standard invasion protocol requires them to set up a command base on a natural satellite, out of reach of the locals, before orbital bombardment can begin. They took one look at our dangerously radioactive moon, and decided it would be easier to invade someone else.
As Samuel said, there would be no change to humanity whatsoever. The gamma rays would not penetrate the Earth's atmosphere well enough to affect human DNA. However, depending on the strength of the radiation from the moon, we might see a stronger aurora effect.
I think what would be more notable is the effect on our society's development. Let's assume the world developed the same as it did without the radioactivity. Consider these two scenarios:
Since so much radiation has to come from something, this means there could be a valuable radioisotope in the dust of the moon. NASA has been using decaying radioisotopes for long term power generation for decades. The prospect of unlimited energy has made the world more excited than ever before! Now capitalism has jump-started a gold rush to the valuable materials of the cosmos, causing massive investment in space exploration!
Conversely, society has scorned space travel! In the Space Race with the USSR, the United States exposed all those Apollo Astronauts to radiation, killing them a few years after the missions ended. As a result, Human Space Exploration has been banned because of the inherent risk. With NASA now gone, anti-science rhetoric is at an all time high. The boundary between church and state has begun to thin...
As jedmeyer mentioned, there would likely be no direct effects from the radiation. The earth already gets hit with a lot of high frequency EM radiation, including gamma radiation, and very little of it gets through the atmosphere.
There would likely be quite a few secondary effects though:
- Earth would have been warmer throughout it's history. All the extra radiation would be adding extra energy to the atmosphere, which has to go somewhere. How much warmer is hard to tell without exact numbers on the composition of the moon. This in turn, would likely have at least resulted in shorter ice ages, which would have had an impact on both evolution, and human society.
- The exact atmospheric composition would likely be different, though by how much is hard to tell. It's well known that UV radiation is what produces almost all of the atmospheric ozone on Earth. Similar reactions occur for gamma radiation, though to a much lesser degree.
- On the same note, lightning might be more common in the upper atmosphere. Ionizing radiation like gamma radiation can cause charge imbalan es that are conducive to creating lightning. Such effects occur relatively infrequently however (thermonuclear explosions for example can cause lightning, but it's mostly because of the atmospheric turbulence and charged particulates, not the radiation directly).
- Man-made satellites would have lots of issues. Gamma radiation is a serious problem for electronics. It's possible to design systems that are radiation hardened, but they're far more expensive than regular designs. Most LEO satellites are not fully radiation hardened in real life, but in a world like you describe, they would have to be.
- Human spaceflight will become almost impossible, with becoming an astronaut essentially being a death scentence.
- The moon might look brighter in the night sky.
- Baseline background radiation on the surface might be marginally higher (a few PPM at most). Anything that can be done to impacted by this would be, but the effects would be so small they likely would be unnoticeable except on a massive scale.
That last part is where most of the theoretical potential benefits come in. Bacteria and other things that already have a particularly high mutation rate would likely evolve faster and scientists would have another natural random number generator. That's about it though as far as benefits I can think of.
One non-benefit: A seriously radioactive moon would put a big crimp in our space program. Just about everything that's been tried in deep space has been tried on the Moon first. If the moon is hot enough to bathe Earth in gamma rays it's hot enough to play major havoc with the electronics of any nearby spacecraft--you probably can't even orbit it.
You weren't very clear on whether "humanity's inception" was an inclusive or exclusive timeframe, but either way it presents a problem since it means the moon has either been outputting a large amount of energy over billions of years or somehow got "turned on" once humanity appeared.
Still, the point is that this isn't natural at all and contradicts all current theories of moon formation. Therefore those theories are obviously wrong and the moon is giant spaceship parked there by ancient astronauts.
The radiation might just be the first layer of automated defenses that will fall under the control of whoever first reaches the bridge of the Lunar class starship, gains administrative rights over the ship's systems, and inherits the legacy of the Precursors. Or if you insist on the radiation itself being helpful, it might be part of an electronic countermeasures package designed to protect Earth from detection by roving bands of von Neumann machines bent on devouring all organic life.
Now, if the radiation is coming from right below the surface from a claimable source, like radioactive isotopes, building a base on the moon might be more feasible. However, this depends on the amount that is radiated. Too high and there will be no space program, faint but detectable would work perfectly. We could have had project orion if the moon contained claimable nuclear material.
The periodicity that the moon brings via gravitational radiation (i.e. tides) creates a dynamic playground of wet/dryness that add complexity to the environment. This complexity is exploited in various ways by quite a few different organisms (consider a bird hunting in tidal pools, or a barnacle advancing further upshore).
If the moon emitted gamma radiation, which acts as a DNA mutagen, life on earth would have a similarly periodic landscape regarding the rate of mutation that occurs during meiosis (which is when parent DNA combines to form child DNA).
This would allow organisms to vary their behavior and in turn expect more or less variation in offspring. During times of dense population, or for parents that cannot offer a socioeconomic advantage, it might make sense to procreate when the moon is overhead--in hopes that the offspring benefits from the 1:1,000,000 shot that it will receive a beneficial mutation.
On the other hand, during times of sparse population, or in cases where the parents can offer an advantaged situation, it would be better to "play it safe" and procreate when the moon is not visible. This would minimize mutation generally and be culturally a move that indicates support for the status quo.
Realistically, I think that the x-ray moon children are just more likely to just be sickly--genetic evolution is not so potent on human timescales--but as a cultural phenomenon it might lead to an even stronger conservative/liberal divide than we have today.
It gives lifeforms an elevated tolerance to radiation.
Since the entire biology on Earth developed under moderate radiation it adapted to that radiation by optimizing our DNA mechanisms. As an benefit, humans (and other species) are able to withstand high radiation levels without lasting damages.
Advantages include fields as manned spaceflight, healthcare (less cancer) and nuclear industries. The main detriment is that governments have less scruple in using nuclear weapons.