Washington D.C. is a great city. It is not a highly concentrated federal cluster surrounded by "ghetto" neighborhoods like previously mentioned.
First of all, the city is very dense by American standards. It's very walkable and you can get around by subway or bus relatively easy. However, in comparison to NYC, greater Washington D.C. is not nearly as dense, nor as walkable. For instance, much of SE is underserved by the subway. NYC also has a lot more mixed use construction. The SE quadrant is soon getting a street car system that will improve it's connectability greatly. I will mention a larger chunk, percentage wise, of D.C. contains undesirable areas then NYC at this time. Still, there are PLENTY of great neighborhoods and gentrification is is changing neighborhoods much faster in D.C. due to it's smaller geographical size and new positive emerging reputation.
Diversity, describes both cities well. In both you will find an assortment of people from all over the United States and abroad. I would say that you will meet many more people with advanced degrees in Washington D.C. then NYC (Per capita). It's noticeable, NYC has a wider socioeconomic and educational spectrum. I am pretty confident that the NYC and DC metropolitan areas are the two most diverse in the USA.
The D.C. metropolitan area cannot truly be described as conservative, however it is much more so then NYC. You don't see nearly as many as the odd going ons you would in NYC. However, D.C. is become increasingly liberal with the growing population of young professionals and diversity. I would rate the city Moderate, with a liberal edge. NYC by comparison I would rate highly liberal.
Energy wise, it's a slower paced city. Do not mistake that for being at a standstill because there is plenty to do, and a variation of jobs operating at different times of day. However most weeknights the place is pretty dead by 3am. Weekends about 4am. You won't see nearly as many musicians and performances, nor as many going ons throughout the city as NYC.
The city also seems more interdependent on it's suburbs then NYC. Most New Yorkers seldom leave the five boroughs, or even their own.
To those who compared it to Brooklyn, aesthetically, somewhat I agree (at least the more core primarily residential neighborhoods of DC like U St, Dupont, Shaw, Adams Morgan, Columbia Heights, Capital Hill). However, Brooklyn is much more fast paced, dense, and liberal.