I live in the District of Columbia, known locally as “DC”, and worldwide as the location of Washington, the capital of the United States.
Outside DC, people tend to use the terms interchangeably, but here, we like to make a distinction:
“DC” or “the District” is a city where people live, work, play, and generally go about their business just like any other large city in the United States or the world. Many people who live in DC actually have nothing at all more to do with the federal government than people living anywhere else. DC is not a state, but has a local government that performs most of the functions of a state government as well as a local one.
“Washington” is the “federal city”, the seat of government, the location of Congress, the White House, the Supreme Court, etc. The federal government, with all of the bureaucracy and corruption and general inhumanity people associate with it, is conducted from Washington, not DC. When you elect people to go to the White House or Congress and they let you down, you sent them to Washington, not DC. DC doesn’t even get to send people to Washington other than one non-voting “delegate” to the House and zero Senators. (We do get to vote for President, though.)
“The DMV” is a recently popular term for “the District, Maryland, and Virginia”, i.e., DC plus the surrounding parts of the other two states. People say this instead of something like “the DC Metro”, because around here, “Metro” refers to the subway.
So remember, when you’re complaining about the government in Washington, don’t blame DC. DC doesn’t like Washington any more than you do; in fact, most of us like it even less because of the countless ways Washington screws us over.
(Some of you will say “well, actually, DC is the Federal District provided for in the Constitution which contains the city of Washington blah blah blah…” You are technically correct, the worst kind of correct. You win nothing. Please don’t say DC when you mean Washington. Thanks.)