Correlation is not a good measure of dependence when the underlying variables have non-elliptical distributions. It is also not a good measure for all elliptical distributions, for example, when returns have a multivariate normal distribution. It is a good measure for some (but not all) elliptical distributions.
If returns are not normally distributed, then a zero correlations does not necessarily imply that returns are independent.
Correlation is not invariant to transformations so for example, the correlation of two random variables A and B will not be the same as the correlation of ln(A) and ln(B).
