In general, t-test is a parametric test of the difference in mean between two groups that assumes that the data are normally distributed and the groups have equal variances. Welch's approximation can be used when the assumption of equal variances is not satisfied. While it is commonly believed that the WMW (Wilcoxon-Mann-Whitney) test compares medians in the same way that a t-test compares means, this does not always hold. In some cases, a WMW test indicates a significant difference between groups when the medians are equal, in contrast to what many researchers would expect. There are two versions of the WMW test: a general form and the location-shift model, both of which can be problematic when skewed discrete data are analyzed.

I hope you did not get confused ideas.

I concur with the previous comment. The decision is whether or not your data meet the assumptions of the parametric test (t test) - if they do not, then you need to use the non-parametric alternative (Mann Whitney U). If you are using paired data, you will alternatively need to use the Wilcoxon Signed Rank test. 

See:

http://vassarstats.net/utest.html

http://vassarstats.net/wilcoxon.html

Thank you both! I was able to solve my issue thanks to your help!

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With zero inflation a single test is not appropriate. There must be a factor driving the zero inflation. Are your data counts (intiger values)? This would point you in the direction of zero inflated poisson regression, for example.

But for sure, neither the t-test nor the WM-W test fits with a zero-inflated data generation process.

The t-test, actually, is incredibly robust. So much for the folklore about its assumptions. It does not, repeat not, assume that the data are normally distributed. Please read up on the assumptions elsewhere, but don't take answers on RG as anything more than folklore being repeated from teacher to student without any data to back them up.

But, as I said, the t-test does not fit with your data because there are two underlying processes, which need to be tested with a zero-inflated model.

Thank you Ronán, I'll keep looking into it. My data are not integer, they are counts of events divided by elapsed time. For the treatment group, the event count is usually zero, so as a result the event/time is zero.

I don't believe that the zeros arise from a different process to the non-zeros. Would that mean a poisson regression would still be appropriate?

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Aha - then they are counts. You need a Poisson model, which allows you to specify both the count and the duration.

If there is an excess of zeros, they are coming from somewhere, and regardless of your belief there is no reason to assume that they somehow just happened by accident. One reason can be that groups with different means are mixed in the data. For sure, you will need robust variance estimation. What software have you got?

You could use a probit test on a binary variable.

In SAS using proc glimmix:

if count = '0' then y=0; else y=1;

model y = trt / dist=binary ;