is there a functional difference between a western tie down and an english standing martingale?
Functionally, not really. The main difference is that a western bridle usually lacks a noseband, so the noseband is a part of the tie down, whereas the english standing martingale loops around the bottom of the noseband which is already part of the bridle.
Western ^ with noseband as a part of the tie down, not part of the headstall.
English ^ where the noseband is already on the bridle, so the tie down/standing martingale just loops around that.
There’s also more diversity in western tie downs. Some of them are more or less harsh depending on the material and thickness of the noseband.
^ This is going to be fairly gentle on the nose because it’s broad and flat.
^ This is gonna hurt.