Compare random samples and make inferences about difference in distributions I have a general question about comparing independent samples of continuous data and making inferences about difference in distributions. I'm really struggling with which approach to use.
Besides of tests on location parameters, like expected values or medians, and dispersion, are there other approaches?
How about using correlation coefficient? Does a high correlation necessarily indicate no difference in distributions?
And what are the drawbacks of these approaches? Like tests of difference between means require normality assumption?
 A: 
Does a high correlation necessarily indicate no difference in distributions?

No.
Imagine you have paired values that are from the same distribution and which happen to be highly correlated (say the correlation is almost 1). Let's say they're both standard normal for example. Now divide the second set by 10 and subtract 10.
The two distributions are now very different, but their correlation is unchanged.
(Or instead, multiply the second set by -1. Now the correlation is extremely low, but the distributions are actually the same!)
Note that the pairing would be unusual -- normally you'll have two independent samples you want to compare distributions of.

Besides of tests on location parameters, like expected values or medians, and dispersion, are there other approaches?

You can compare almost any aspect of a distribution you choose, it depends on what you're interested in finding out (what alternatives you're most interested in being able to pick up). There are thousands of tests that have been described and many more than are possible if you have a need for them; it's not feasible to list them all. You could compare upper quartiles. You could compare some measure of skewness. There's almost no limit to what might be done, under some set of assumptions and some desired comparison.
There are also general tests of whether two parent distributions differ -- i.e. for a null of identical distributions against an alternative that the distributions are different (keeping in mind that failure to reject doesn't imply that the distributions are actually identical, only that we couldn't distinguish them).
The best known of these is the Kolmogorov-Smirnov two sample test (sometimes just called the Smirnov test).
If you're interested in general tendency for values form one distribution to be larger or smaller than the other (not just pure location shift), you might consider the Wilcoxon-Mann-Whitney test. But there are other things you could test; it really depends on what matters for your situation.

And what are the drawbacks of these approaches? 

Well they all carry assumptions which may or may not be reasonable for your case; you need to consider the suitability of and impact of violations of those assumptions. Some tests can be badly affected by fairly modest violations of the distributional assumptions, others might be fairly robust to them, but may still be sensitive to violations of assumptions of independence, for example, or perhaps assumptions about equality of variance.
Additionally any test you choose will have poor power against some situations and (hopefully at least) good power against other situations. You need to choose your tests keeping in mind where you most want to have good power.

Like tests of difference between means require normality assumption?

Well, no, not every test of difference in means requires a normality assumption.
Firstly I can make some other parametric assumption and may be able to construct a suitable test for equality of means. 
Secondly, I could use (for example) a permutation test of means which doesn't require any specific distributional assumption (the test is distribution-free). Or I might use bootstrap testing.
Thirdly, I could add some assumptions to a nonparametric test  (say a Wilcoxon-Mann-Whitney test) which if those assumptions hold would be sufficient to make it a test of means. 
