Used to, ABHE was considered the undergraduate equivalent of ATS. Since 2004, however, ABHE has started granting accreditation to Graduate institutions, so I am not sure how that has changed things.
I would probably say that as undergrad work goes, they are probably equivalent to ATS (although ATS does not accredit undergrad inst.). At the Grad level, they are certainly better than TRACS, but not as prestigious as ATS...probably...
I think the accreditation list, for organizations that accredit theological schools, probably looks like this...
#1 Regional Acc.
#2 ATS
#3 ABHE
#4 DETC
#5 TRACS
Again, this is mostly a guess; accreditation discussions is one of my hobbies.