In REG or GLM, x is the independent variable and y is the dependent variable. The model tests the hypothesis that the x variable is linearly predictive of the y variable.
This can be looked at as a test of linear independence, but it is not a test of general independence; it's not sensitive to some non-linear relationships in the data.
CORR will look at monotonic relationships (in the Spearman or Kendall's statistics), but it will still miss a "bathtub" relationship.
So, do graphs to look at your data too.
Doc Muhlbaier
Duke