How do you find the difference in distance between earth and Mars at different points in their orbits?
There are several ways of calculating the changing distance between Earth and Mars.
The simplest way of calculating the distance between Earth and mars is to use their orbital parameters. These describe the shape and orientation of planets' orbits.
To get the distance, first of all calculate the angle between the two planets relative to the Sun for each point. This is the difference between the longitudes of the two planets. Then calculate the distance of Earth and Mars from the Sun.
The distance between Earth and Mars can then be calculated using the cosine rule.
The orbital parameters assume that the orbits are elliptical. In reality the orbits are constantly changing. The NASA DE430 data enables the calculation of the positions of planets to high accuracy. In particular it makes it easy to calculate the distance between any two of the Sun; Moon and planets for a given time.