BBE and Aphex are both used in post-production and the mastering process by many professional acts. I find that which one does better varies and I think it depends on which process was favored in the studio - if any was used at all. I tend to use BBE more than Aphex - simply because it is more available on apps than Aphex. Right now, some Cowan players, older versions of Jet Audio, JVC mobile player and BBE player use the BBE. Aphex is only available on the Aphex player. However, with both BBE and Aphex, less is more and using that little of each makes both sound very similar. Cranked all the way up, both sound terrible and horribly V-shaped.
BBE is shifting higher frequencies with lower amplitude ahead in time (actually it's shifting lower frequencies behind) and Aphex is adding in even harmonics. Aphex does better at playing back vintage music recordings which are dull sounding due to missing harmonics - since it actually adds them into the recording - BBE doesn't do well with these because the upper harmonics are not there to shift.
I see BBE outboard boxes available on the vintage market more than Aphex.
There are some who think these boxes sound just the same as Eq - and for those people, they are not stuck like I am in that I can only listen to music rigs with an outboard box or specific apps.
I stuck a BBE box into the PA at the club I DJ at.
One thing I've come to realize is that Behringer's attempt to do this thing falls short of both. The Behringer Dualfex tried to copy BBE and Aphex and stick them into one box and uses a knob to balance between the two. While it does brighten the music, it doesn't sound as enjoyable as the original effects. I have one of these Behringer boxes at the bar also - which also stuck limitjng/compression, noise reduction, and exciter into one box. I've turned off all the effects in the Behringer except the limiter.