"Emotional problems" is more subjective, has no established meaning, where as "mental illness" has diagnoses certifying them as a mental illness.

I agree that it is not "just" emotional problems; depression, bipolar disorder, schizophrenia, etc. usually have emotional problems along with the illness, but having emotional problems does not guarantee the opposite, that you have a mental illness.

I don't see it as an insult, just an indication of ignorance on the part of anyone who would define mental illness as "emotional problems"; but I've never known anyone who thought/expressed the concept of mental illness that way.