As far as I know, the applications are for theoretical problems and has no application in everyday work.
The simplest example would be testing how close f is to a linear function.
We consider f:{0,1}^n->{0,1}. f is linear if f(x+y)=f(x)+f(y), and there are different ways to measure almost linearity. Either the probability that randomly picking x and y, we have f(x+y)=f(x)+f(y), the other would be how far away the function is from a linear function in hamming distance.
Using the tools in analysis of boolean functions, we can show these two measures are basically the same.