In short, a track warrant is simply a train order which has much of the verbiage pre-printed on the form, thus making it easier to copy by a crew member on a moving train. Perhaps having the order pass through fewer hands reduced the potential for errors, but it certainly meant that many of those "hands" were out of a job.
The other kind of "track warrant" is issued at a crew's starting point as a cover sheet for the track bulletins pertaining to their territory, and conveys no authority in and of itself to occupy a main track. In other words, it is an update of the time-honored "clearance card."