Testing difference between two means. Skewed ( N1=21) vs Symmetric bell (N2=47). Wilcoxon rank sum test appropriate? I need to test to compare the means of two samples, one with size 21 and the other with size 47.  Histograms show Sample 1 is skewed to the right while Sample 2 has a bell symmetrical shape. Descriptive statistics will be found below. I tested for normality with Shapiro-Wilk any way, rejecting the null hypothesis for either, so that will confirm they are not normal. I understand t-test will not be appropriate to this case. I moved to non-parametric and used Wilcoxon rank sum test and got p-value 0.2395 which will indicate there is no evidence to think the means are different (they are equal). 
Is it correct to use Wilcoxon rank sum in this case where the shape of both samples is different? Am I missing something? Or with this information can I state that there is no difference between the means? Is it necessary to do something else like permutations? 
Some statistics...
Sample 1
Shapiro-Wilk p-value = 0.000
average = 0.0270
sd = 0.0892
kurtosis 10.4
skewness 3.2
n = 21

Sample 2
Shapiro-Wilk p-value = 0.0366
average = 0.0367
sd = 0.0752
kurtosis 0
skewness 0.6
n = 46

 A: There are several problems with what you have written:

I need to test to compare the mean of two datasets: one with 21
  samples and the other with 47. Histogram shows Dataset 1 is skewed to
  the right when Dataset 2 has a bell symmetrical shape. Descriptive
  statistics will be found below. I tested for normality with
  Shapiro-Wilks anyway not rejecting the null hypothesis for any of
  them, so that will confirm they are not normal.

If you failed to reject the null in SW then you have not confirmed that they are not normal; that would be the conclusion if you had rejected the null.

I understand t-test will not be appropriate to this case. 

The t-test is reasonably robust to violations of normality if the variances are not too different and the sample sizes not too different

I moved to non-parametric and used Wilcoxon rank sum test and got p-value 0.2395
  which will indicate there is no evidence to think the means are
  different (they are equal)

The Wilcoxon test does not exactly compare means, it compares mean ranks.
A p-value of 0.24 does not indicate that there is no evidence to think the mean ranks are different; it indicates that there is not enough evidence to conclude they are different at whatever level of significane. Three is, however, some evidence that they are different.

Is it correct to use Wilcoxon rank sum in this case where the shape of
  both datasets look different? Am I missing something? or with this
  information, Can I state that there is no difference in the means? Is
  it necessary to do something else like permutations? I'll really
  appreciate your advice. Thank you.

You cannot state there is no difference in the mean ranks; you can state that there is not sufficient evidence to conclude they are different. 
If you want to compare mean ranks, then what you have done is fine (if you  correct your conclusions) but if you want to compare something else, you will need a different test; maybe permutations. 
