Is Washington state ans Washington D.C. 2 different names for the same place?

No, they are 2 different places. One as you said is a state- Washington. Located in the northwestern part of US. It borders Canada to the north Oregon ( a state ) to south and the State Of Idaho to the east. To the west is the Pacific ocean.
Washington DC is a City. It is on the east coast of the US. It is the Capitol of the United States, Its where the White House is. The DC stands for District of Columbia.
On TV Washington DC is talked about a lot , esp when it comes to government talk and it is often called simply Washington.