Why is $\{0,1\}^n$ referred to as the Boolean hypercube?
Strictly speaking, the "Boolean hypercube" is not the same as the set $\{0,1\}^n$. The Boolean hypercube is the graph whose vertices are the set $\{0,1\}^n$, and whose edges are defined as follows: two strings are adjacent if and only if they differ on exactly one bit.

I guess that usually, people refer to $\{0,1\}^n$ as the Boolean hypercube when they really want to think of it as a graph and not as a set of strings.