(1) Re-sampling is certainly a possibility, as suggested in a previous answer. Also. possibly a permutation test or a bootstrap confidence interval, which are different kinds of resampling than suggested there. (If you can figure out how to
do one of these, this might be your best bet.)
(2) If there are few (if any) ties within or between the two samples, a Mann-Whitney test (equivalent to a Wilcoxon rank sum test) would be a possibility. However, if there are many ties, such rank-based tests lose power and require adjustments.
(3) Because you have relatively large sample sizes, unless the populations are far from normal, you can use a t-test (the Welch 'separate variances' version would be best) for approximate results. The degrees of freedom will probably be large, so the distribution of the T statistic will be close to normal. By 'approximate results', I mean that the P-value will be only approximate, so I would feel
uncomfortable rejecting the null hypothesis unless the P-value is less than 1% or so.
Notes: (a) The Welch T-statistic essentially provides the answer to your original
question. For your sample sizes it should be nearly normal and
you will reject that the two population means are the same if the absolute value of the T-statistic exceeds 2 (nominal 5% level).
(b) Most statistical software does the 'Welch' two-sample t test by default, doing the 'pooled' test only if you insist.)
(c) If you want to leave a note with more information about the nature
of your data, maybe I can give some advice about what will work best in your particular case.
Addendum: As an illustration of the behavior of markedly nonnormal distributions, I simulated 100 observations from the strongly skewed distribution Gamma(4, rate=1/4), which has population mean 16, SD 8. My sample had $\bar X = 15.78$ and $S_X = 8.07$. I also generated 50 observations from the distribution Unif(0, 50), which has population mean 25 and SD 14.43 (and no tails). My sample had $\bar Y = 22.34$ and $S_Y = 13.9.$
A Welch t-test gave T = -3.0762, df = 65.92, and p-value = 0.003.
A Wilcoxon signed rank test gave p-value = 0.01. Thus both tests convincingly detected the difference in population means between 16 and 25, in spite of rather large and different SDs, and distinct normality of both populations.