The formal names of some illnesses are "X disease". Most of these are named after people -- either the person who discovered the disease, or a well-known person who had the disease.
My cousin developed Lou Gehrig's disease in her early 70s.
List of eponymously-named diseases
As far as I know there are no formal names with "illness". The choices seem to be either "disease" or "syndrome", with an occasional "phenomenon" or "-osis" suffix, or some characteristic of the disease like "anemia", or "cyst" or "palsy".
In general, "illness" and "disease" are synonymous; however "illness" often suggests a more mild disorder. We would ordinarily say someone is "sick" or "ill":
I stayed home from work because I'm sick with the flu.
My family was ill all through the holidays.
If you say someone is "diseased" it sounds pretty serious, as if they have some kind of horrible plague. It's more common in things like scientific journals that talk about the symptoms or infection rates among "diseased persons".
The aim of the study was to assess the occurrence of some putative periodonto-pathogens in "test" and "control" sites in "diseased" and "non-diseased" persons, respectively, from an adult rural Kenyan population exhibiting poor oral hygiene
You can say someone has a disease, but again it can sound better to say they are ill, or use the actual name of the affliction:
A good friend became ill with lung cancer in his early thirties, but so far treatment has kept it in remission.
One of my Facebook friends is sick with leukemia.