I am building an ArcGIS Pro project for Mars. I am trying to compare the difference between two projected coordinate systems Mars 2000 and Mars 2000 (Sphere).
The main difference between the two is the inverse flattening.
As you can see in the screenshot, the Mars 2000 (sphere) does not have a flattening while the Mars 2000 system does.
How does inverse flattening affect the projection and which would be preferred for GIS use?
These aren't "projected" coordinate systems. They are ellipsoidal coordinate systems, designed to precisely map two numbers (a latitude and longitude) to a location on an ellipsoidal model of the planet (where ellipsoidal includes spherical as a special case).
Considering only ellipsoidal geometry for a moment, this means that two points with the same coordinate number values will describe different locations on the two surfaces, so you do have to make sure your data has the correct coordinate system in its metadata. I find it easier to think about an extreme flattened ellipsoid to reason about this, so consider an M+M shape (or Smartie). In this case, the point 45N is very close to the equator, because these are geodetic coordinates, and so 45N is where the angle of the surface is 45 degrees from the N-S pole. On a sphere 45N is half-way.
This will affect distance measurements in the N-S direction. The total distance from the equator (0N) to the pole (90N) is smaller than a similar segment round the equator (0E to 90E) in the flattened model because of the flattening, but also the distance from 0N to 45N is not the same as from 45N to 90N in the flattened model - 0N to 45N is a shorter distance than 45N to the 90N pole (this would be the other way round for geocentric coordinates but lat-long should be geodetic, check the fine details of your coordinate system definition).
Which to use is a matter of application. Assuming Mars is actually more like an ellipsoid with that flattening, and you want distance measurements that need fine accuracy over large areas, then you use the "more correct" ellipsoidal model. The downside is that ellipsoids are more complex shapes than spheres, and so computing distances or projecting to flat coordinates can take longer. So if you are just making illustrative maps then spherical will probably do.
Whichever you use, the important thing is to make sure you know what coordinate system a data set was defined with, and to never change that without transforming the coordinates (or reprojecting a raster) correctly.
Your two views of Mars come from a bijection between the geographic coordinates and a cartesian (XY) axis. This is equivalent to the so-called "Plate carree" projection. In the first case(left), Mars is considered as a perfect sphere. In the second case, it is considered as an ellipsoid (the "radius" at equator is larger than the "radius" at the pole). The second model is more accurate than the first, so you should use the geographic coordinate system based on the ellipsoid (like most of the coordinate systems used for the Earth, the only noticeable exception being the so called "Web Mercator", which is widespread despite the fact that it should be avoided). Just as a rough example, on Earth, the spherical approximation yields approximately 1% error at a latitude of 45° (and basically no difference at the equator).
That being said, for any subsequent GIS analysis, you should use a more appropriate projection on top of your geographic coordinate system, depending on your needs. The errors due to inappropriate projection are indeed much larger than the errors due to the spherical approximation. Just look at the distance between two points at 90° N or S, which should be close to 0. Unfortunately, there are is universally optimal projection, so you need to chose between of conformal projection ("keeping the shapes"), an equidistant projection ("keeping some distances") and an equal area projection ("keeping the areas").
