What is the difference of the earths acceleration measured between when the moon is straight up in the sky...

Question

and when it is all the way the other side of the earth (under your feet)?

Answer 1

Here is a back of the envelop calculation:

The moon has 1/81 the mass of the earth and it's 60 times further away than the center of the earth. Therefore it's gravity is 81*60*60=291600 times weaker. The acceleration due to the earth is 9.813 m/s^2. Divide the two to find 3.4x10^5 m/s^2. The difference between up and down is twice the number.

A small effect indeed, but still enough to generate the tides :)

Answer 2

From the source stated below:

"The gravitational effects of the Moon and the Sun (also the cause of the tides) have a very small effect on the apparent strength of Earth's gravity, depending on their relative positions; typical variations are 2 µm/s² (0.2 mGal) over the course of a day."