As I remember it, when the data is normally distributed then the parametric t-test has a few percentage points higher power (or at least efficiency) than the Mann-Whitney-Wilcoxon (MWW) test. So then they are essentially the same.

But if the distribution is non-normal then for specific distributions MWW can have considerably higher power that the t-test. (To my surprise!)

But MWW is “sensitive to spread”.

So even if two distributions have the same median but have different “spread” e.g. different standard deviation, MWW can give many significances, far more than 5%.

I have heard it been said, although I have nothing written to point to, that the t-test can be better in such situations and “be more non-parametric than MWW”, so to speak.

I seldom use MWW.

I would probably, as Jake suggested above, use a permutation test.

(Or like Englund suggested, try to use a transformation to try to get it more normally distributed.)

(Or expand the potential distributions to the exponential family (where the gamma distribution is similar to the above suggested lognormal distribution) and use a generalized linear model.)

(Or expand it further to the gamlss system...)