Wilcoxon Test

One of the best ways to improve power is by employing repeated measures or matching, and such designs can be analyzed with nonparametric methods. A useful test to see whether the members of a pair differ in size, Wilcoxon’s test resembles the Sign-Test in scope, but it is much more sensitive. In fact, for large numbers it is almost as sensitive as the Student t-test. For small numbers with unknown distributions this test is even more sensitive than the Student t-test.

When we do not know whether values are normally distributed, this test is preferred over the Student t-test.
Comparisons to Other Tests

Ranking discrepancies that can’t be quantified precisely is difficult, which probably accounts for why Wilcoxon’s T is rarely used in that way. The more common use for this test is as an alternative to the matched t test. The Wilcoxon test will usually have as much as 90% of the power of the matched t test; the sign test will have considerably less power.

The Wilcoxon Matched-Pairs Signed-Ranks Test uses the sizes of the differences. The result can differ from that of the Sign-test, which uses the number of + and - signs of the differences.

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Sources:
Essentials of Statistics for the Social and Behavioral Sciences, Barry H. Cohen and R. Brooke Lea (Click for eBook)
http://www.fon.hum.uva.nl/Service/Statistics/Signed_Rank_Test.html