An intersection returns only the items appearing in both lists, discarding anything unique to either.
The question it answers
Where do these two sets agree? That is a different question from what either contains alone, and often a more interesting one.
Typical uses
Keywords both you and a competitor rank for — the terms actually contested. Domains linking to two competitors — the sites that link to companies like yours. Pages appearing in both a crawl and an analytics export — the pages genuinely live and receiving traffic.
Normalise before intersecting
Exact matching applies here as much as anywhere. Inconsistent case or trailing slashes produce an intersection far smaller than the truth.
Try it: List Intersection (Common Lines) on SeoWolf's Notepad.