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 $GENDER$ 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:
$$Y = a + b*X + c*GENDER+ d*GENDER*X $$
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.
The relation between meat type and household budget is of significantly different size.
It could interpreted such that males and females significantly differ in their behaviour considering what kind of meat they purchase depending on their available household budgets.
It is a bit difficult to construct an example, as I do not know how meat type is measured. But, one example supposing that meat type is an ordinal measure of meat quality: A significantly higher correlation coefficient for females could mean that when deciding what quality of meat to purchase, females are more likely to take the houshold budget into consideration, i.e. they buy cheaper meat if the budget is low, whereas for males, if the correlation coefficient is significantly smaller, they are less likely to choose poor meat if their budget is low, or they switch to not that much worse meat if their budget is low.
Best Answer
"Test for equality" is more theoretically correct because you test the (likelihood of) null hypothesis, and it states that the coefficients are equal in the population. But "Test for difference" will be more common way to put it, because it is the alternative hypothesis - the inequality - which a researcher and the reader usually "is after" or has a morbid interest in. Why don't you simply label the table "Correlation coefficients for females and males"?