Trying to choose between these two tests for data I've harvested from Android store. Basically, I want to see if there is any difference in the number of dangerous permissions requested by free vs paid apps. I have equal sample sizes of 1900. When I plot the data they are both highly skewed, almost like decay curves. Under student-t I understand there is an assumption of normal distribution, but not sure what has to be normally distributed, so not sure whether student t would be the right test or whether to use non-parametric mann-whitney?
Skewness will give you trouble with the t-test, yes. You could perhaps do a Mann-whitney, but since the data are counts, you probably need a test that fits with count data.
I'd be inclined to suggest assuming something like Poisson and then conditioning on the sum (giving a binomial test) ... but since you have a mix of applications, there may be additional skewness induced by that heterogeneity.
How skew are the distributions?
How were the applications selected?
You may ultimately be best off treating the applications as a random effect.