The first time these two franchises met in the NBA playoffs fans paid $2-$5 to watch what was then called the Western Division Semifinals, the equivalent of today's first round. The 2015 Eastern Conference Semifinals will be the only fifth playoff meeting between these two franchises in the last fifty years. The first time they were in different cities, the last time one was defending an NBA title, and this time there are only two players (Elton Brand and Paul Pierce) that were alive the last time these two organizations met in the playoffs.
In 1965, the Wizards were the Baltimore Bullets and the Hawks were in St. Louis. In consecutive years, 1965 and 1966, the Bullets and the Hawks faced one another in the first round. The Bullets finished the 1964-65 season 37-43, and the Hawks came in at 45-35. For the Bullets, it was their first playoff appearance. The series began on March 24, and they split the first two games. The turning point in the series was Game 3, an impressive 131-99 beating by Baltimore. All of the Bullets' starters, with the exception of Gus Johnson, scored over 20 points. Guard Don Ohl and center Walt Bellamy led Baltimore with 23 points each. Game 4 was as close as Game 1 but the Bullets finally finished off the Hawks, 109-103, to take the series.
In 1966, the only Western Division (conference) team to finish the season above .500 were the Lakers. The Bullets had home court in the first round due to the fact that their 38 wins were better than St. Louis' paltry 36. The Hawks swept the first two games in Baltimore with their "Iron Man Five" lineup used by coach Richie Guerin. The Bullets were no stranger to team dissention in the mid-1960s, but coach Paul Seymour's sudden revelation that he was stepping down after the season came as a shock right before Game 3. A distracted team entered Game 3 and lost, 121-112. The Hawks swept the series with all of their starters, since they usually played the entirety of games, averaging over 20 points. On the flipside, Don Ohl led the Bullets in a disappointing series with 26.7 points per game. Much of the information about the Bullets in the 1960s and this series are found in Brett L. Abrams and Raphael Mazzone's book, The Bullets, the Wizards and Washington, D.C. Basketball.
The next meeting was a short affair in 1978, and the two teams met for the first time as the Washington Bullets and the Atlanta Hawks. In 1978, division winners received an automatic bye to the semifinal round with the first round being a best of three series between the #3-#6 and #4-#5 teams. The Bullets swept the sixth place Hawks 2-0 on their way to their first and only NBA championship behind franchise players Wes Unseld and Elvin Hayes.
By far the most competitive series between Atlanta and Washington was the 1979 Eastern Conference Semifinals. In 1979, the Bullets were looking to repeat and ran into the fifth-seeded Atlanta Hawks. They split the first two games in Washington and the Bullets won Games 3 and 4 to go up 3-1 in the series. The Hawks stayed alive by winning Game 5 on the road and back in Atlanta they forced a Game 7 with a 104-86 win. Washington coach Dick Motta called Game 6 "the most intense game of the series," even though the Bullets shot a horrible thirty-two percent from the field. According to Abrams and Mazzone, the Hawks got back into the series by out-rebounding the Bullets and with the play of their bench, which outplayed their opponents in the two series-saving wins.
"One of these days, we are going to cut it too close. We are going to lose a seventh game because of injury or a cold performance, and we'll have no one to blame but ourselves." -The Bullets summarizing situation going in to Game 7
Game 7 was on April 29, 1979 in front of a capacity crowd in D.C., which included then President Jimmy Carter who sat with Bullets owner Abe Pollen. The final game was as tightly contested as the rest of the series. The Atlanta guards, Eddie Johnson, Armond Hill, and Terry Furlow made life difficult for Washington's backcourt but Elvin Hayes and Bobby Dandridge picked up the slack and dominated the Hawks frontcourt. Hayes scored 39 and Dandridge finished with 29, which included a dominating fourth quarter of 17 points. The Bullets also had 30 points from offensive rebounds that propelled them of a 100-94 win and a trip to the Eastern Conference Finals. The Bullets were just thankful they had survived. "It's too bad they're [Hawks] not in the west," said Elvin Hayes. "They deserve to go further."
The level of competition in the 1979 Eastern Conference Semifinals surprised some, was appreciated by most, and proved that "NBA players were not so blase and jaded that they are apathetic about their art." Thirty-six years later, the Atlanta Hawks (at least since they have been in Atlanta) have yet to make it out of the second round of the playoffs, though they were in the Western Division Finals in 1970, and the Wizards have not made the conference finals since the Carter presidency. For the Hawks, a win puts them in a place the organization has never been, while the Wizards can go where they haven't been in a long time.
Washington and Atlanta playoff stats here via Basketball Reference.