What is the difference between disjoint union and union?

The notation $A\sqcup B$ (and phrase "disjoint union") has (at least) two different meanings. The first is the meaning you suggest: a union that happens to be disjoint. That is, $A\sqcup B$ is identical to $A\cup B$, but you're only allowed to write $A\sqcup B$ if $A$ and $B$ are disjoint.

The second meaning is that $A\sqcup B$ is a union of sets that look like $A$ and $B$ but have been forced to be disjoint. There are many ways of defining this precisely; for instance, you could define $A\sqcup B= A\times\{0\}\cup B\times \{1\}$. This construction can also be described as the coproduct of $A$ and $B$ in the category of sets.

(This ambiguity is similar to the ambiguity between "internal" and "external" direct sums; see for instance my answer here.)