This is a good question. When I saw the movie for the first time, this threw me off, but when I watched it a second time I saw the power of it.
The type of film a director uses make a huge difference. Part of the reason for this is because the form of something has certain connotations. In this sense, the form of something is a conveyor or meaning, just like the content.
So, in the case of the film, Schindler's List, the black and white film brings the viewer of the film back into the time period of World War II, where black and white was what filmmakers used in that time period. For this reason, it gives a sense that we were actually there. Also it gives the film a very documentary feeling, in my opinion. This, too, has certain connotations, namely, that what you are watching is not merely a movie for entertainment, but a documentary for education.