Vinegar will sometimes work. A product called CLR works really well.
Fill a baggie with vinegar and secure it so that the head of the faucet is in the vinegar, let it soak over night. If you can take the tip off and the screen, soak them in a dish overnight. This should take care of the faucet. The counter top should be cleaned according to what the counter top is made of, some don't like things like vinegar and could cause problems. If you know what they are made of, research what cleaners can be used on your counter top.
Vinegar and CLR do both work. CLR works faster. I just use a tooth brush and scrub around the faucet. Once it is all removed I make sure I wash the area well to get the CLR off of the counter.