It is reasonable to use the Wilcoxon-Mann-Whitney test as you have done. There are alternative models you could have used --- such as negative binomial regression for count data, or ordinal regression --- but the results would be similar.
This may be an example where the p value doesn't tell you what you need to know. Because your sample size is very large, the low p value indicates that one can tease out a "signal" against the "noise" in the data. Here, the W value is very large and this results in a very small p value. That's all good and reasonable.
But the next questions are, How large is the effect size? Is this a meaningful difference? I have some R code below to start to sketch out answers to these questions.
### Install packages, read data, prepare data
Data1 = read.table(header=T, text="
Group Mentions Count
A 0 13249
A 1 1565
A 2 292
A 3 65
A 4 19
A 5 6
A 6 4
A 7 3
A 8 2
A 9 1
Data2 = read.table(header=T, text="
Group Mentions Count
B 0 5043
B 1 348
B 2 66
B 3 18
B 4 8
B 5 3
Data = rbind(Data1, Data2)
Long = Data[rep(row.names(Data), Data$Count), c("Group", "Mentions")]
rownames(Long) = seq(1:nrow(Long))
We could plot histogram-like bar plots to see the relative distributions of the mentions for each group. Due to the distribution of the data here, these plots may not be very helpful.
Long$Mentions.f = factor(Long$Mentions)
histogram(~ Mentions.f | Group, data=Long, layout=c(1,2), col="darkgray")
It may be more useful to look at the proportions of each level of mentions for each group. Here, Group A has about 87% zero mentions and Group B has about 92% zero mentions. Is this an important difference? That's for you to say.
Table = xtabs(Count ~ Group + Mentions, data=Data)
### Group 0 1 2 3 4 5 6 7 8 9
### A 0.87 0.10 0.02 0.00 0.00 0.00 0.00 0.00 0.00 0.00
### B 0.92 0.06 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00
And my WMW test agrees with your results.
wilcox.test(Mentions ~ Group, data=Long)
### Wilcoxon rank sum test with continuity correction
### W = 43702925, p-value < 2.2e-16
For a more formal effect size statistic, we might look at Cliff's delta. This value is related to the probability that an observation from one group would be larger than an observation from another group. Here, the value 0.05 is quite small, and the positive value indicates that the first group ("A") is slightly stochastically dominant.
cliffDelta(Mentions ~ Group, data=Long)
Vargha and Delaney's A gives us the same information in a different form. Here the answer is literally the probability that an observation from one group would be larger than an observation from another group. The answer is about 52%.
vda(Mentions ~ Group, data=Long)
The upshot is that your use of the WMW test is reasonable. But the large sample size gives the test power to detect small stochastic differences between the two groups. It's important to also assess how large these differences are, and if these differences are practically meaningful for you.