Actually its not quite that simple. The approaches are "equivalent" (in terms of operational/statistical predictions) but not "isomorphic" (even if we restrict the regular linear algebra approach to the same gates as I use in the book). Basically (because of not implementing normalization/unitarity) in certain situations the "mist" retains a count of the number of "Feynman paths" that led to the output state. For example, concatenate two PETE boxes (hadamard gates) and the output is [W,W], concatenate four and its [W,W,W,W]. Operationally identical (you only observe the ball to be certainly white!) and applying a suitable simplification rule you "drop the extras" of course. But if someone (unlike me!) wants to think of these states as having ontological status maybe they would think the distinction is relevant.

A much more interesting example of something similar happens in the Deutch-Hayden version of Heisenberg picture style quantum theory. There from looking at the "state of the world" you can determine which total unitary evolution occurred to get you there. In the Schroedinger picture, if I give you the inital and final states there are many possible unitaries that get you between them. Its possible to go further and come up with versions where not only the total unitary connecting initial and final state is discernable, but also the sequence of Hamiltonian's that implemented it are too.

These things are mainly of interest to people in foundations of course (since they like to play the game of blurring the line between our math and what is "really out there"). I tried to be very careful in the book to maintain the distinction.