Fundementally different from "Chubby", the adjective "Fat" refers to a health condition in which the number of fat cells within the body exceeds the body's natural carrying capacity for lipids.
A fat body will exibit signs of overbearence of weight; such as the formation of rolls and cellulite, as well as a host of other adverse phisiological symptoms.
Contrastingly, "chubby" is a body type describing a body that has a naturally high carrying capacity for fat cells, and is unrelated to the aforementioned health condition. However, a chubby body can also become fat if the number of fat cells in the body surpasses the body's natural carrying capacity.
As such; a person may be fat, chubby, both, or neither.
"I'm not fat, I'm just a bit chubby!"
"I can see the rolls through your shirt man, you're fat."
"I think I'm a bit of both!"