They are closely related. One of the main motivations behind the notion of a set (or more generally of a class) is as an object that corresponds to a logical predicate: the set/class is an aggregation whose elements are precisely the objects satisfying the predicate.
If $P$ is a unary predicate, and I use the notation $[P]$ to denote the class corresponding to $P$: i.e. the class satisfying
$$x \in [P] \Leftrightarrow P(x)$$
then we have
$$[P \wedge Q] = [P] \cap [Q]$$
so we see the close relationship between $\wedge$ and $\cap$.
But for sets $S$ and $T$, $S \wedge T$ doesn't really make sense. Notation like this is would appear, however, if you were working in a lattice whose elements are sets: in this case, $\wedge$ is not meant to be viewed as an operation on sets, but as an operation on lattice elements (which just happen to be sets).