Why does earth not follow the same rule around sun as moon follows around earth? As you all know that only one side of the moon faces earth through the whole of its journey. The reason why it happens can be owed to the fact that the center of gravity of the moon (because of gravitation of earth) and center of mass of the moon doesn't coincide because of difference in the relative distance of it's far and near side from the earth. So as a result, the near side always faces the earth.
But if we apply the same theory to the Sun and the Earth, why is it that only one side of the Earth doesn't face the Sun every time as the Earth's center of mass and center of gravity too would be different. 
 A: The center of mass IS an object's gravitational center. The moon is tidally locked to the Earth because tidal forces slow an object's rotation. This will eventually stop the object's rotation. The moon is actually slowing the Earth's rotation due to Earth's tidal energy loss, but it is at a  very slow rate, the Sun will likely become a red dwarf before the Earth is tidally locked to the moon.  The Earth is not tidally locked to the Sun because solar tides are very small compared to lunar tides, so Earth looses even less rotational energy to solar tides than to lunar tides. You might want to see https://en.wikipedia.org/wiki/Tidal_locking
A: Welcome to StackExchange.  Good question. The moon rotates at the same rate at it revolves around the earth. This is caused by tidal forces. 
From here I quote: We know that tides here on Earth are caused by dragging of the Moon. In the reverse, the Earth is also dragging the Moon with its gravity, creating bulges on two sides of the Moon. If the Moon does not rotate and revolve at the same period, then the bulges will no longer point directly to and against the Earth. The gravitation force on the two bulges will then become uneven and create torque to rotate the same side of the Moon back to the Earth. The “locked” phase is the most stable geometry for the Earth-Moon system. Just like rolling a ball in a valley, even the ball starts to be on a hill, it will be dragged to the bottom of the valley eventually by gravity.
This same effect is acting on the earth and it may become tidal locked many (billions) years into the future. This can be seen from this question on this site. And as mentioned in the comments.
