I tend to think that the biggest difference is socially constructed. For one, things like cancer are easy to objectively determine. Either you have a malignant tumor or your don't, clearly you're either ill or you're not. But it's easier for people to just invent new mental illnesses and have it seem legit, for example how homosexuality was medicalized. Similarly, the false distinction allows for perpetrating stigmas about mental illness. It's not the same as physical illness, it's not real, it's all in your head, just get over yourself sweetie.
I have severe, chronic depression. I'm currently in the process of applying for disability resources because I am disabled by my illness. And one of my biggest fears there is those around me will condemn me as some lazy person wanting things handed to them or whatever. Most reasonable people wouldn't similarly condemn a person confined to a wheelchair for not walking up the stairs.