Do Fuses make a difference ib sound?

No, not that you would be able to reliably discern by listening. In addition, the directional aspect is pure BS.  At least I have never heard a difference and I have tried many different fuses.  As I posted many years ago, if you want to reduce noise and improve the contact of your fuses, wrap the glass cylinder with a couple of passes of teflon tape and, assuming chassis mounted fuse holders, use small orthodontic rubber bands around the two prongs connecting each end of the fuse in order to secure the contact.

The Digital Fuse Box is basically a magnetic breaker.  Magnetic breakers are now being used instead of power switches by one of the equipment manufacturers of my gear.  I put copper slugs in my monoblocks and installed two wall-mounted magnetic breakers to protect them (a couple of pics on my system page).  I can’t say they sound any better, but they certainly do not sound any worse.  One breaker protected one of the amps one time so I know they work.

They make a huge difference. Pull out the power fuse, and there’s no sound.

Hafler didn’t think so.  All of my Haflers have the output fused. Those has saved my butt while my head was up it.   The difference between with and without can’t be discerned.  Keep em. 

Can someone explain the mechanism(s) that create the differences that are attributed to different brands of fuses?

Expectation bias is a strong force...Bussman and Littelfuse products may be the only fuses you need, and serious equipment designers  like Nelson Pass agree. However, if you can imagine a magical property and apply it to an already working fuse design, and charge multiple times it's actual cost to make a pile of money for yourself, do it!  Some people will feel better about themselves by thinking it's money well spent and apply magical results in their little minds. Years ago I got a pile of Synergistic Research fuses from a friend and carefully tested 'em changing direction, etc., recorded the differences between them and standard fuses (of which there were none), and sent them back. Enough for me!