Logistic regression and Wilcoxon test I ran a multivariate logistic regression with glm in R with some continuous and some categorical variables. Only continuous variable $A$ showed a p-value of < 0.05 and a confidence interval which did not stradle 1.  
Running a Wilcoxon test (actually a Mann-Whitney test because the samples are not paired) with $A$ divided into the two outcomes groups returns a p-value of 0.15. This indicates that there is no difference between the means of $A$ in the two groups.  
How do I reconcile these two results? The logistic regression result indicates that $A$ is a predictor of the outcome, but the Wilcoxon/Mann-Whitney indicates that there is no difference between the two groups.
 A: Did you just fit one big glm model then look at the individual p-values?
Remember that those p-values are measuring the effect of that variable above and beyond all other variables in the model.  It is possible that more of your covariates are really contributing, but there is redundant information, so they don't show significance.  It could be the combination of A along with another covariate or 2 that shows the real difference, but A by itself is not meaningful.
Also look at the effect sizes and standard errors for all terms.  There is a paradox associated with the Wald tests/estimates (the individual p-values in the standard summary) that you can have a very important variable appear to be non-significant because the standard error is over estimated.  Likelihood ratio tests are much better because of this.
You can assess both the above by fitting a reduced model using only A, then use the anova function to compare the 2 models, if that is significant then it indicates that there is something important in your model beyond the A variable.
A: The tests make different assumptions, and so do not give exactly the same result. The bigger problem is the (incorrect) assumption that failure to reject the null "indicates that there is no difference". It does not. It just means that you don't have enough evidence to reject the null of no difference.
A: The more appropriate comparator is the proportional odds model, which contains the Wilcoxon test as a special case.  And note that 'multivariate' refers to the simultaneous analysis of more than one dependent variable.  I think you meant to say 'multivariable'.
A: If I understand things correctly, you're testing if the mean value of predictor $A$ associated with outcome $1$ differ from the mean value of predictor A associated with outcome $2$. Even if they don't differ, this result says nothing to your research question. What it says is only that in your sample, the average value of the predictor $A$ is'n different among succes and failures of the dependent variable. I'll give an example so you can see why.
Imagine that i'm interesting to know if education affects the probability of an employee being manager in a given firm. I collect a random sample of employees, with $200$ employes that are not managers and $70$ which are actually manager. I ran a logistic regression and the effect of education is significant (usual caveats about p-value apply here as well). 
Now, you may wonder if the sample is balanced, which is, if there is the same variation in education for managers and non-managers. One first test to see this is to test if their means differ. If their means don't differ, than you know that on average on your sample, managers and non-managers have the same level of education. You could check if there is roughly the same variation of educational level among managers and non-managers. But I think that it's pretty clear that the Wilcoxon test (or the Mann-Whitney) has nothing to say about the effect of your predicton on the probability of success (in my example, success means to be a manager).
A: The p-value that you got for the parametric test (from glm) depends on the effect size, i.e., by how much can the predicted values change depending on the value of A. Now if a small proportion (that proportion may be dependent on other predictors) of predicted values change by a large quantity due to the variable A, you have a large effect size and significant correlation and significant p-value. The rank-sum test, however only cares about the effect size in as much as it changes the ranking. If you were to push all of the overall highest 10 values (regardless of their class) by a large number in the absolute scale so that the rankings remain the same, the rank sum statistic will not change while parametric tests will. If you have  reason to believe the effect size or the distribution you assumed (probably normal), then use the p-value from the parametric test, otherwise use rank-sum.  
