They are two seperate software programs, if that's what you mean. I think they use a different underlying library to do the actual mp3 decoding, but as far as you can tell on the CLI the differences are mostly cosmetic:
$ mpg123 amonTobin-catPeople.mp3
High Performance MPEG 1.0/2.0/2.5 Audio Player for Layer 1, 2 and 3.
Version 0.59s-r9 (2000/Oct/27). Written and copyrights by Michael Hipp.
Uses code from various people. See 'README' for more!
THIS SOFTWARE COMES WITH ABSOLUTELY NO WARRANTY! USE AT YOUR OWN RISK!
Title : cat people Artist: amon tobin
Album : Year :
Comment: Genre: phreakout tunes
Playing MPEG stream from amonTobin-catPeople.mp3 ...
Found new ID3 Header
MPEG 1.0 layer III, 160 kbit/s, 44100 Hz stereo
$ madplay amonTobin-catPeople.mp3
MPEG Audio Decoder 0.15.2 (beta) - Copyright (C) 2000-2004 Robert Leslie et al.
Title: cat people
Artist: amon tobin
Genre: phreakout tunes
As you can see, mpg123 is a little more verbose by default. Is there a reason you cannot just fire them up and see for yourself?