It has significant impact on several issues relating to a FE.
Consider the options, we can have a far sun. If this is the case the apparent size of the sun will remain constant, and the direction to the sun will be the same all over Earth, in Cartesian coordinates.
This means if Earth is flat, everyone would see the sun in the same apparent direction. It would rise at the same time for everyone and set at the same time for everyone, and be overhead at the same time for everyone.
However if Earth is round, the apparent direction will be based upon the angle of the ground, if the sun is overhead for someone it will be below someone 180 degrees around Earth.
So if the sun is far away, then Earth must be round.
However if we have a near sun, then as you move around Earth the angle to the sun varies, as does the apparent size.