The official tourist season starts from April through September but if comfort is a priority then the best time to visit the US capital is in Spring ( March to May) and Fall (September to November). A word of advice is to buy advance tickets to popular attractions beforehand because the queues can be very long.
Summers are hot and humid (July and August) but accommodation rates fall during this season due to slump in business travel.
If you want to see snow then winter is the time for you! Keep in mind harsh winters and snow storms sometimes shut down the city, particularly in January.