The faster Earth's (compared to Moon) rotation smooths the average heat. The higher planet’s surface specific heat (oceanic waters vs dry regolith), also smooths the average heat. Consequently Earth's daytime surface temperature (compared to Moon) lessens, and Earth's nighttime surface temperature (compared to Moon) rises.
Moon receives 30% more solar energy (per unit area) from sun than Earth – because of the big difference in albedo. Earth's albedo aearth = 0,306 and Moon's albedo amoon = 0,11 only.
Thus, what is left (after the albedo losses) to "absorb" is for Earth (1 - 0,306)So = 0,694So and for Moon is (1 - 0,11)So = 0,89So.
Moon /Earth = 0,89So /0,694So = 1,282 or 28,2% higher = 30% higher
But there is also something else very interesting happens.
What makes Earth on average a warmer than Moon planet is the difference between Earth’s and Moon’s emitting temperatures.
Because of Earth's 29,5 faster rotation and because of the Earth's surface mostly being wet (the watery planet) the daytime Earth’s surface is warmed at a much lower temperatures and therefore at the daytime Earth’s surface emits IR radiation at a much lower temperatures. So the intensity of Earth’s daytime IR radiation compared to Moon is much lower.
It is much-much lower than the expected 30% difference (because of albedo).
So there is a great amount of energy – compared to Moon – “stored” on Earth during the daytime emission...
This “saved” energy should be emitted during the nighttime then.
At the nighttime Earth’s surface is warmer (than Moon's) and therefore Earth’s surface emits at the nighttime IR radiation at a higher temperatures (than Moon's). So the intensity of Earth’s nighttime IR radiation is higher.
There is always a balance
The energy in = the energy out
But there is something else very interesting happens.
In order to achieve that balance Earth’s nighttime IR emitting intensity should be much higher than the nighttime IR emitting intensity of the Moon.
Now we should take notice of the nonlinearity of the Stefan-Boltzmann emission law. Consequently the nighttime temperatures on Earth rise higher (compared to Moon) than the daytime temperatures on Earth lessens.
So the average Earth’s surface temperature is warmer (compared to the Moon).
Thus Earth’s Tmean.earth = 288 K and Moon’s Tmean.moon = 220 K
The faster rotation and the higher specific heat does not make sun to put more energy in the Earth’s surface.
What the faster rotation and the higher specific heat do is to modify the way Earth’s surface emits IR EM energy, compared to Moon, (per unit area).
Earth emits IR radiation at lower temperatures during the daytime and at higher temperatures at nighttime. Because of the nonlinearity of this process according to the Stefan-Boltzmann emission law Earth ends up to have on average warmer surface than Moon.
The nighttime temperatures on Earth rise higher (compared to Moon) than the daytime temperatures on Earth lessens.
Earth receives 30% less than Moon amount of solar energy. But this energy is “more welcomed” on Earth, so there are more energy accumulated in the Earth's case.
To illustrate the above conclusions I’ll try to demonstrate on the Earth-Moon temperatures comparison rough example:
Moon…100 K...220 K …390 K
Earth…184K↑↑.288 K↑.330 K↓
So we shall have for Earth, compared to the Moon:
..........Tmin↑↑→ T↑mean ← T↓max
+Δ84°C↑↑→ +Δ68°C↑mean ← -Δ60°C↓
The faster a planet rotates (n2>n1) the higher is the planet’s average (mean) temperature T↑mean.
To emphasize we should mention that Moon’s max and min temperatures are measured on Moon’s equator, and Earth’s max and min temperatures are not. Earth’s max and min temperatures are measured on continents, and not on oceanic waters.
Otherwise the Δmin would have been even bigger and the Δmax would have been much smaller.
This rough example nevertheless illustrates that for the faster rotating and covered with water (higher cp) Earth compared with Moon the average temperature should be higher. The planet’s faster rotation and the planet’s higher specific heat “cp” not only smooths, but also processes ( Δmin > Δmax ), the incoming solar heat, but in a different emission pattern.