Hassidic Jews are a subset of Orthodox Jews. They are more insular and traditional. There are many different Hassidic groups, each one hailing from a different village in Eastern Europe (pre-Holocaust.) They retain many of the customs and dress from that place and time period. Each group has it's own religious leader, called a "Rebbe."
Black Hat Orthodox, known among Orthodox Jews as "Yeshivish," are usually not Hassidic, but they may be. They wear hats and jackets out of respect for God. They think that just as one who is standing before a human King would be very careful to be dressed appropriately, so too one who is standing before God, the King of Kings.
In terms of actual Jewish Practice, there is no fundamental differences between these groups. They are all Orthodox Jewish, and there are only minor differences, usually about different customs. All follow the Halakha, Jewish Law. However, Hassidim do have a more insular world-view than other Jews, and view Western Popular Culture as "bad."