Give your large sample sizes, you could probably use a t-test on the means. If your sample sizes are equal, you are in pretty good shape whether you want to use a pooled estimate of the variance or unpooled (Welch's test). Do a one sided test, if you are sure that the population of s1 has a mean at least as large as the mean of the population of s2.
Note: If the variances are much larger than the means, your counts are not Poisson. But what matters here is the distribution of the sample averages, and that should be nearly normal, unless the data are super-skewed. In that case, you could do a non-parametric test like the Kruskal-Wallis.