Are the order finding and period finding algorithms the same thing?

Do they both just use similar methods of calculation, or are they completely interchangeable?

If you look at Nielsen and Chuang, figure 5.5, page 241 (at least in my 2002 version), order finding and period finding are separated. (Note that this is talking about the problem definition, not the algorithm for solving them). Essentially, period finding aims to find the least positive $$r$$ such that $$f(x+r)=f(x)$$ for all $$x$$, while order finding is doing a special case of this where the function is of a specific form: $$f(x)=a^x\text{ mod }N$$.