Is adrenal insufficiency the same as addison's disease?

Not exactly: Addison's disease is one very specific cause of adrenal insufficiency, in which the immune system makes antibodies directed at adrenal proteins which end up destroying the adrenal glands. There are many other causes of adrenal insufficiency including bleeding or infection of the adrenals glands, destruction from cancers that have spread there, or failure of the pituitary to stimulate the adrenals.

Not exactly: Addison' s disease is a state of adrenal insufficiency but not vice versa as there are many other causes besides ad ( an autoimmune disease) that can lead to cortisol deficiency state such as tuberculosis of the adrenals, hemorrhage of the adrenals, certain genetic storage syndromes and hypopituitarism.