Do you want to show that the (linear) relationship between a variable A and a variable B is stronger for one group than for another? If the answer is yes, then I would go for a regression analysis.
Suppose that there are two continuous variables $X$ and $Y$, and a group variable called $\rm GENDER$ that is equal to 1 for men and equal to 0 for women. I would like to know if the relation between $X$ and $Y$ is different for men than for women. I would then run the following linear regression model:
$$Y = a + b\times X + c\times \text{GENDER}+ d\times\text{GENDER}*X $$
I would then test the joint hypothesis if the coefficients $c$ and $d$ are equal to zero. If this hypothesis is rejected, then I would conclude that there are differences between men and women.
What do you think?
EDIT: I will add some more explanation, as requested, but I am afraid that this will take us too far away rom the original question ...
I have taken the original problem, and I have tried to give it some structure. More precisely, I have modeled the continuous variable Y as a function of another continuous variable X and a discrete variable called GENDER. The functional form chose here is a linear one. The lower case letters represent the parameters of the line.
In fact, the above equation looks like one line, but it contains two: one for men and one for women. The parameter a is the intercept for women, (a+c) is the intercept for men, b is the slope for women, and (b+d) is he slope for men.
The parameters c and d mirror the gender differences, or more generally speaking the differences between the two groups. I have used gender for illustrative purposes, but you can replace it by what you want: color, species, marital status, ... Thus, if these two parameters (c and d) are simultaneously equal to zero, there is no (apparent) difference between the two groups, and thus the relation between X and Y is the same for the two groups.
