there are different format of image exist. in the application of object tracking, we generally apply background subtraction then left out image is binary in nature. so it is one application can be said. likewise , how many applications are there?
There are so many applications of binary images. One of these for image pre-processing. Also, in mathematical morphology, there are several kinds of binary images.
I imagine most optical measurement applications (lenght, diameter, area etc.) end-up with a binary image at some point - reducimg the measurement task to simply counting pixels.
Edge detection and similar algorithms will follow a similar processing path, as well as optical character recognition.
Thus, the binary image representation is the central point of a multitude of image evaluation algorithms.