There are differences.
Fading black will fade all over, or in the usual 'burn' areas - the butt on either side of the tail is usually the first to go IME. Fading black is still genetically black, and can be homozygous. There was a theory that fading was hetero and non fading was homo but that was disproved. So a fading black horse would be aa for agouti. This horse is fading black.
Bay is your 'normal' bay, with a generally even tone. By that I mean the colour stays the same to the 'soft' parts of the body - muzzle, flank, elbow etc. Bay is caused by an A on the agouti locus - so a bay horse is Aa, AAt or AA. This is your average bay.
Brown is a bay horse with lighter 'soft' parts. They can be light bay to start with, or dark bay, even all the way to almost black. The soft parts of the horse are places that would not be the first to fade, and would not fade to the extreme seen in brown. Brown is on the agouti locus, and is At. A brown horse is Ata or AtAt.
This horse is brown. With a really dark coat, you can clearly see the soft points and how light they are. These points could not get this light without some other bleaching on the horse.
The confusion most people seem to have is that brown can look a heck of a lot like bay, just with the lighter soft points. Brown can still have black legs, even with really defined stockings. It can be on a really light horse. This horse is tested as Ata.