Thank you all for pointing out the issues! As for the formulas: Variance of a sample has different scaling factor, hence the difference. Use the second CentralMoment instead:

In[175]:= data = {1, 2, 3};

In[176]:= wd = WeightedData[data, {w1, w2, w3}];

In[177]:= Variance[data]

Out[177]= 1

In[180]:= Variance[wd]

Out[180]= (w1 w2 + 4 w1 w3 + w2 w3)/(w1 + w2 + w3)^2

In[178]:= Variance[wd] /. {w1 -> 1, w2 -> 1, w3 -> 1}

Out[178]= 2/3

In[179]:= Variance[data] (Length[data] - 1)/Length[data]

Out[179]= 2/3

In[181]:= CentralMoment[data, 2]

Out[181]= 2/3

In[183]:= CentralMoment[wd, 2] == Variance[wd]

Out[183]= True

It's good that you pointed this out and it would be helpful if the specific documentation errors were described. For example the documentation for Mean with weighted data shows

The $\frac{1}{n}$ should not be in that definition.

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 *)

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

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

Variance[EmpiricalDistribution@data]

This is documented in Mean[], Variance[], and StandardDeviation[] under "Details":

The data can have the following additional forms and interpretations:... WeightedData — weighted mean, based on the underlying EmpiricalDistribution  »

(This comes a couple of items after the erroneous formulae that have been pointed out already.)