The first place I would look at when trying to get some details about the Math behind the implementation would be the references mentioned in the documentation for the classify function.
Wikipedia also provides some basic information about Quadratic Discriminant Analysis(the method that is chosen when the 'type' input to the classify function is 'quadratic') as seen here.
The Mahalanobis distance and its application in discriminant analysis is also talked about in this Wikipedia page. However, I would rely on the references mentioned in the documentation page for the details.
With respect to one of the options working better than the other, from experience, that depends completely on the data that you use.
