Repeated Measures vs. Randomized Groups

Can someone please explain to me why tests of significance are more powerful if they can use correlated repeated measures rather than randomized groups?
suppose that i have a group of depressed patients, i measure their depression on scale for example, Beck depression inventory. then i gave them a medication and remeasure their depression. Any change of the depression score before and after will be considered significant and will be attributed to the medication.
Now consider another situation, when i compare the depression score of two groups, one without medication and the other on medication. The difference between two groups may be due to the medication given or may be due to difference already existed between the two gruops. So, a large difference is needed to produce significance.
A powerful test is the test able to detect any small change, elicit significance.
i hope it will help