For two particles to influence each other you need some sort of interaction.
For (macroscopic) mass this is clearly Coulomb-interaction. Two atoms can not be at the same place, because their cores repell each other. If you look at smaller scales, strong and weak interaction might add their part.
Photons have no charge, no color-charge and don't interact weakly. So there is no way for them to interact directly with each other. It might be that gravitation has an effect, but one can easily assume that it is negligible. (A picture of two people standing on different sides of a soccer-field, shooting with a gun at a 90° angle to each other comes to my mind. Then assume the bullets don't hit each other but only attract each other via gravitation and calculate the effect. Might not be in any way a properly scaled example, but it should give an idea).
However if you go even deeper into particle physics, you will see, that photons indeed can interact with each other via higher order processes. See e.g. this website.
As far as i remember from last semesters particle physics lecture, photon-photon-scattering has indeed been observed, but the cross-section (basically: "how often does this happen?") is quite a few orders of magnitude to low to actually see the effect with your eyes.
(Think of the two people with guns at the soccer-field again - how likely is it that their bullets actually hit each other?)