Adrenaline vs epinephrine
Solution 1:
"adrenaline" and "epinephrine" are synonyms, and both words are currently used in medicine, anywhere in the world.
In medical contexts, in the U.S., you are likely to hear "epinephrine" more often than adrenaline.
In non-medical contexts, phrases like "adrenaline-charged", "adrenaline-junkie", "adrenaline rush", are "fixed" and I've never heard "epinephrine" being substituted.
Solution 2:
Epinephrine is a specific catecholamine, one of a whole list of adrenergic drugs.
http://www.healthline.com/health/adrenergic-drugs
Adreneline is a non-specific term that could refer to epinephrine in particular, but also (especially outside of medical context) to any positive chronotrope (agent that increases heart rate).
Solution 3:
Indeed, Adrenaline was at first a trade name, and like many trade names got picked up by the popular culture and became slang. Scientists and doctors are usually more careful to avoid adopting a trade name in this manner; hence, "epinephrine" to refer to the hormone produced naturally in the body. These synonyms sound so different because one is built from the Latin "ad" (above) and "renal" (of the kidney, as in "renal failure") + "ine" (generic suffix for a substance, particularly a drug) and the other is based on the equivalent Greek "epi-" (on top of, as in epicenter, epidermis, or epidemic) and "nephr-(os)" (kidney, as in nephrologist--a kidney doctor) + "ine". The adrenal glands are so called because they sit on top of each kidney.