The simple answer is:
The difference between a parrot and a parakeet is that a parrot is a more like a broad term for the whole parrot family encompassing parakeets and parrots respectively; and parakeet are actually a small, specific member of the parrot family.
Where does the parrot and parakeet names come from?
The word ‘parrot; is believed to be derived from the Old French ‘perroquet’, which has two Rs.
Whereas, the word ‘parakeet’ supposedly has its root in the old Spanish word ‘periquito’ for parrot or there is another potential origin in the old French word called ‘paroquet’. Most people spell 'parrot' with two Rs, and 'parakeet' with just one, though sometimes people use 'parrakeet' with two Rs.