I got it now. I don't understand why both expressions are useful, but I see that they're the same.
Exactly. There's really no reason to use one or the other. The two-equation form is the canonical representation, but the equations can be written as a single equation, which sometimes appears in textbooks, too. The order of calculating the IIR part and the FIR part can be exchanged. But that's all just computations. It's not really useful to have the distinction unless you're actually programming the filter, and trying to optimize your code. It's enough to see that there are a few different formulations of the same thing.
A few other functions that you can make with these are integrators and differentiators. Not just simple ones, we can actually make integrators/differentiators of higher precision, by using several of them. Since all polynomials over the real number field can be decomposed into first/second order factors, we can build filters of arbitrary order (if we know how to factor them). Beyond a certain point, it becomes more efficient to use fft rather than biquads.