The designs of the bass and the pickup are virtually identical, and the woods - and fit and finish - used on the NXT are superior. That could actually be an initial drawback if you're aiming for authenticity; since the NS basses are tightly constructed and are solid-body instruments, their habits lean towards a more electric sounding bass (more sustain, more string detail, that sort of thing.) Better woods, better construction might enhance that sustain, whereas the goopier poly finish and the less stringent QC on the WAV might provide what some consider to be a less "brash" sound. Just a theory.
Find out what strings are on the WAV. If they're the original strings, they're an exceedingly cheap set of Chinese-made strings (I actually still have some stock replacement sets, believe it or not!) that I don't like any more than I like the Contemporary strings which come stock on the NXT (which, to my ear, are too bright and "electric" sounding.) But the strings on the WAV, if they're original, are now several years old and have been "played in" to a point where they're more darkened up (not so "zingy" as new) which will probably get a more pleasant sound to many players.
Additionally, the player you borrowed the bass from may have made adjustments to the instrument, allowing more "upright-style" technique. NS Design sets up the NXT basses before they are shipped to dealers, and they set them up with VERY low action. Too low, IMHO. (It's probably to bolster the confidence of the electric bassists who try the bass when it's on display at your local guitar superstore.) To get that "big" URB sound, the strings need enough clearance to be plucked with vigor - and enough room to "bloom" without buzzing against the fingerboard. I raised the action a little on my WAV4, and that alone improved its sound quite a bit, because I could PLAY it more like a real URB. Maybe that's what your friend did as well?
Just a few thoughts to consider...