Because of chemistry, my friend.
Mars is red because it has a lot of iron on its surface, and that iron likes to react with oxygen.
When iron and oxygen get together, they form iron oxide, also known as rust. Rust is red, or at least red-orange, depending on how much oxygen it has.
Contrarily, the moon's white colour is a result of its low oxygen and iron content.
It is mostly made of silicates, which are minerals that contain silicon and oxygen. In general, silicates can be either white or grey in colour, depending on their reflectivity.
Mars cooled down more quickly and lacked a powerful magnetic field to shield it from the solar wind because of its lower size compared to Earth.
The solar wind is a stream of charged particles that comes from the sun and can strip away a planet's atmosphere.
The Martian atmosphere was nearly destroyed when this object wiped it off the face of the planet. Mars has lost a lot of its oxygen-rich water and carbon dioxide.
However, there was still a significant amount of iron present in its crust and mantle, which were exposed to the residual oxygen in the air and rocks.
The iron gradually turned red as it oxidised.
The moon is even smaller than Mars, so it cooled down even faster and never had an atmosphere or a magnetic field to begin with.
It also formed from debris that was ejected when a gargantuan object hit the Earth eons ago.
That debris was mostly silicates, with very little iron or other metals.
The moon also didn't have much volcanic activity or water to bring up more metals or oxygen from its interior. So it stayed gray or white.