Which state is farther west?

For practical purposes, Maine is the furthest east state in the U.S.

However, in a technical sense, the western portion of Alaska's Aleutian Islands are the furthest east part of the U.S., because they extend past the 180 degree meridian, halfway around the world from the zero meridian that runs through Greenwich, England. "East" and "west" in longitude coordinates on a map or GPS are calculated according to which direction is the shortest distance from Greenwich.

As an example, Fairbanks, Alaska is at longitude 147.7 W, a little less than 5/12ths around the world heading west from Greenwich. If you continued heading west from there to Attu Island in the Aleutians, you would pass the halfway mark of 6/12ths, and so the location would be measured from Greenwich heading east instead. Attu Island is longitude 173.2° E.