Actually, we live in Alexandria, Virginia which is across the Potomac River but it is in the Washington metropolitan area. If traveling overseas, an Australian from the Sydney suburb of Auburn would tell people they met that they lived in Sydney knowing that the listener would not be familiar with Auburn but it seems that most Americans, at least older ones, make the distinction and would say "Alexandria near Washington" or "Lakewood near Denver". I don't know if this has always been the case or is a result of the "white flight" that left most American city centers poor and black.
However, in the case of Alexandria, the difference probably is significant since Alexandria and Washington were on opposing sides in the Civil War. The state capital Richmond is just 150km/90 miles was the capital of the slave-holding Confederacy, the "South" while Washington was the captital of the abolitionist "North".
In any case, I'll have lots more to say about Washington but you already know it is the center of world power and that there are lots of things to see in the area. More about Alexandria and Washington.