Newtonian physics allows different frames of reference. Non inertial frames of reference need pseudo-forces. Einstein's relativity deals with non absolute space and time, that is, time speeds up or slows down depending on your 'frame of reference', the only thing remaining constant is the speed of light in a vaccum.

The choice isn't purely one of convenience because assuming 'heliocentrism' allows for a more elegant and universal set of equations to explain everything, from quarks to galaxies.