Variance of WeightedData gives the population variance, while Variance on a set of raw observations gives the unbiased sample variance.

I also think there is a problem with WeightedData and the documentation... With the same data generation:

Mean@d
Mean[WeightedData[d, 
  w]] (*Correct, but different from the documentation of Mean!*)
Mean[d w]/Total[w] (*From the documentation of Mean for WeightedData *)

As fo the variance, the problem seems linked with unbiasedness :

Variance@d  (*unbiased*)
Mean[(d - Mean@d)^2] (nbpoints/(nbpoints - 1))(*unbiased*)
Variance[
 WeightedData[d, 
  w]]  (*  Problematic definition in the documentation, \
for  WeightedData*)
Mean[(d - Mean@d)^2]  (* biased *)

To add to David's point, data will be treated as population data if wrapped in EmpiricalDistribution[]:

Variance[EmpiricalDistribution@data]