Both methods are equivalent, nothing strange that they gave identical result.
is just another version of H-H equation - note that right side equals zero for half titration. Try to understand how to rearrange HH to obtain this version.
It will not work in case of a very weak acid, but for benzoic is OK.
So, your error in both cases has the same theoretical source (neglectable) and identical experimental problems. Plus, you probably ignored ionic strength
of the solution.