Equivalence of Mann-Whitney U-test and t-test on ranks What does it mean when people say that a t-test performed on ranked data is equivalent to a Mann-Whitney U-test? Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values? The reason I ask is I tried both in R and compared two groups with very small sample sizes (3 and 4). I got completely different answers: one significant and one not.
The two groups are A=(1,2,3) and B=(4,5,6,7).
t-test: p = 0.01
Mann-Whitney U-test: p = 0.06
 A: 
Does that mean they just test the same hypothesis/are useful in the same situations or are they are supposed to give the exact same p-values?

It means: 
(i) the test statistics will be monotonic transformations of each other.
(ii) they give the same p-values, if you work out the p-values correctly.
Your problem is that the t-test on ranks doesn't have the distribution you use when you use t-tables (though in large samples they'll be close). You need to calculate its true distribution to correctly calculate the p-value. It matters most for small samples ...  but they're also the ones where you can calculate the actual distribution most easily.
A: N.B. Not really an answer per se...
Jonas Kristoffer Lindeløv wrote a recent blog post here that addresses relationships between linear models and group tests (such as t-test, Mann-Whitney, Wilcox, Kruskal-Wallis, ANOVA, etc.). He also created a somewhat limited simulation to assess the differences between the t-test and MW directly.
It would be nice to either have 1) an analytic calculation or 2) a strong set of simulations to create some good rules-of-thumb for when we can use a linear model on ranks as opposed to a nonparametric test, as these are sometimes more convenient.
