Actually, stripping off or adding an electron doesn't do anything to the number of neutrons. All it does is give you a charged atom.
Why is that important?
Because when you shoot a stream of charged atoms close to a magnet, the magnet will pull them off course. When you shoot a stream of uncharged atoms close to a magnet, nothing happens.
It is the magnet pulling them off course that makes the mass spec work. Lighter atoms get pulled off course further than heavier atoms, so if you shoot a stream of charged atoms containing a mixture of C12 and C14 close to a magnet, it will separate into two streams, one of C12 and one of C14. Then your detector can measure how many atoms are in each stream, and tell you the relative proportions of the two atoms.