Is the test statistic useful beyond finding your p-value? If the p-value suggests significance, can the test statistic give any more information? I.e., can you use the test statistic to tell if some of the significance is more significant than other parts?
I ran Mann-Whitney U tests and these U's are, overall, far higher than any other U's in my analysis, even for other tests that resulted in p<.001. Is it valid to say that the test which resulted in the 8.00 or -7.66, for instance, were more significant than others, given that no other U statistic of the 48 tests I ran even exceeded 5.00?

Edit: In response to Glen_b's comment, I relooked at my output and sure enough Mann-Whitney U is a separate value from the (standardized) test statistic.

I suppose, then, my question hasn't changed, because I'm still asking about the test statistic, just properly labelled now. What am I supposed to call those values in my write-up?
 A: It probably won't be helpful for most readers to report the W or U statistics.
The "standardized test statistic" appears to be the z value, or z statistic.  This is helpful to report, as most readers will be familiar with it, and will have some sense of what the values mean.
In a sense, the value of the z statistic could be interpreted as "more significant" or "less significant".  Though, if you are reporting p-values to as low as < 0.001, making any argument for differences in p-values lower than this probably doesn't carry a lot of meaning.  Simply reporting the z-values and the p-values probably conveys the information you want to convey.
Reporting effect sizes will probably be meaningful for your purpose.  For Wilcoxon-Mann-Whitney test, a relevant effect size statistic expresses the probability of an observation in one group being larger than an observation in the other group. There are a few statistics that express this, including: Vargha and Delaney’s A, Glass rank biserial correlation coefficient, Cliff’s delta, Grissom and Kim's Probability of Superiority.
You can also report other meaningful comparisons, like the differences in medians.
As to practical significance --- I like to use the term practical importance to avoid confusion --- this is really a judgement based on your field and specific circumstances, and can include things like economic considerations.  It might be partially based on the p-value or the effect size.
