It's not really as much of a difference as you're implying. The distinction here is whether the other party intended to cause you harm. We overlook minor, commonplace things like catching a cold from someone for two reasons; first, it's rarely deliberate, and second, we all expect to accidentally give someone else a cold or similar at some point, so on the whole it evens out and isn't worth getting upset about. However, if someone deliberately exposed you to a disease, however minor, that would be a form of assault, just as if they'd beat you up or broken your leg, or injected you with poison.

It's worth noting that "illness" can refer to conditions which are not contagious; cancer, for example, or malnutrition, or a chemical imbalance. Merriam-Webster gives the medical definition as "an unhealthy condition of body or mind". It's a very general term. While mental conditions certainly do not seem to resemble viral or bacterial infections (unless you count certain memes...), one can hardly argue that they aren't "unhealthy conditions".

I would say the term illness is more applicable, the the general sense, than injury, though injury (abuse) may be involved in causing the illness. A blow to the head is an injury; the resulting concussion is an illness. A similar relationship applies between mental abuse and mental illness.