I've been reading the first Harry Potter book, and as one character's "tip of the nose" was referred to as 鼻先, another character's was 鼻の頭.
Is there a difference between the two? When would one use one or the other?
There is actually a difference between the two.
「鼻{はな}の頭{あたま}」 can only refer to the "physical" tip of one's nose -- nothing more. Therefore, it has a highly limited meaning.
「鼻先{はなさき}」 can refer exactly to what 「鼻の頭」 does and something extra. It can also refer to the larger (empty) space in front of one's nose, eyes, face and even one's body.
Thus, you can have acne both at 「鼻先」 and 「鼻の頭」.
「鼻先に/鼻の頭にニキビができた。」 (ニキビ = acne)
But a car can whizz by only at your 「鼻先」.
「車{くるま}が鼻先をかすめて通{とお}った。」 (かすめて通る = to whizz by)