I use both. There is no question that JB Bore cleaner is the fastest most effective way to get copper out of a bore. I don't know yet about the Bore Bright stuff. There are many people that will say you don't want the bore "too shiny" or too smooth. I shoot and barrel a lot of rifles and I have a borescope. I cant really see where the Bore Bright stuff hurts anything, and I cant really say that it necessarily helps either. It is such a fine polish that it will not take out tight spots or tooling marks. As others have suggested, too smooth of a bore finish will promote copper fouling, yet I don't find this to be true either. It didn't make or stop any of the rifles I tried it on foul and it didn't alter accuracy in any way either. Because I saw no change, other than the obviously very shiny bore when you hold it up to a light and look thru, I kind of quit using it. It would seem like a very high polish in the bore would make for easier cleaning, but I didn't really find that either.
I have lead lapped a few barrels and used several different grades of lapping compound...while I did get results, I cant see, with my borescope much difference in the finish appearance of the bore surface at all in regards to the grit size used. That is to say that 220 grit lapped in the bore looks very little different than 320 or even 400. Lead lapping will take out tight spots and it will smooth out some light tooling marks, it will also make the bore oversize if you go too far. I doubt you could ever change much with some polish on a patch no matter how much you rubbed on it.