Why do both the VIF and tolerance statistics exist, when the latter is just the reciprocal of the former? Is taking the reciprocal helpful in some way, or is it just a matter of historical accident that there came to be two terms to describe the same thing?

(From the Wikipedia page.)  
 A: They both exist as mathematical objects because you can take the reciprocal.  I take it your question is: "why do we have names for both, and include them in classes / textbooks?"  Here are some thoughts:  


*

*Tolerance: This is very intuitive, and it is easy to see where it comes from and what it means.  Moreover, in a regression class, the tolerance and VIF are always taught together, with the tolerance coming first (I usually discuss these topics in the following order: pairwise correlations $\rightarrow$ $R_j^2$ $\rightarrow$ tolerance $\rightarrow$ VIF).  It serves as part of the walk in 'you have to be able to walk before you can run'.  

*VIF:  The variance inflation factor is somewhat more useful interpretively.  That is because the VIF tells you how much larger the variance of your sampling distribution is than it would have been if the variable had been orthogonal to all other variables in your dataset.  For example, if the VIF for variable $x_j$ is $9$, then the ${\rm SE}(\hat\beta_j)$ is $3\!\times$ larger than it would be if $x_j$ were perfectly uncorrelated with your other variables.

A: The tolerance has a straightforward interpretation as the unique variance accounted for by a regressor. 
But the VIF has the neat property that because it is an inverse, it is much more sensitive when tolerance is close to zero - which is the point where you should be worried about unstable estimates due to multicollinearity.
