yes, the question is a bit vague. Also on first sight it also appears trivial.
I would like to test if x1=x2. That means I have two samples, meaning two vectors x1 and x2. Now I want to know if x1(1) = x2(1) , x1(2) = x2(2) , ... , x1(end) = x2(end).
Sounds easy, in octave I simply write x1 == x2
The trouble is, that in reality x1 is not _exactly_ equal to x2. In reality it is more like x1 = x2+random noise. So the question could be asked like this: how much noise is allowed before the null-hypothesis x1==x2 should be replaced by the alternative hypthesis.
Correlation answers to "how precisely do my samples match the equation x1 = b*x2+a"
I would like to know "how precisely to my samples match the equation x1 = 1*x2+0"
Correlation would give the same answer for x1=x2 and x2=5*x2 , so it cannot tell the difference between two equal sample and to highly correlated ones.
The whole question revolves around simulation models. I would like to have some meaningful answer whether my model works. And to me it seemed obvious to ask if measured values are equal to simulated values, for the same set of independent variables. So x1 are measured values, x2 are results of the simulation, and the simulation works if the simulated values are statistically significantly equal to the measure values.