Both methods use different mechanisms as to how they work (and everything works for 6 weeks)
Body by Science focuses on Sarcoplasmic growth - the mitochondrial dense "goo" that surrounds the muscle fibers, some of the muscle fibers themselves will grow as well.
StrongFirst focus on the nervous system - neural mylination, synaptic facilitation and myofibralar density - the body's nerve wire shielding, "circuit efficiency" and the side effect is the muscle fibers themselves grow in numbers, also creating a bit more sarcoplasmic "goo."
A sarcoplasm emphasized program will produce rapid (yet much more temporary) size gains and it's not uncommon to see strength gains. It works, but not forever. However, if you were to hop onto one for a quick 2-4 week cycle between traditional strength cycles you could really help yourself out because of all the extra mitochondria (and therefore ATP) you'd get from the extra sarcoplasm. In addition, these programs tend to increase capillary density (blood vessels, yo) in the target muscles so your recovery ability will improve.
I've tried both methodologies of training and sided with StrongFirst because I like the "practice" aspect of it and ..good luck finding a competitive lifter be it power, oly, or strongman that lifts to failure and has even an entry level poundage let alone a record breaking total. "Reverse engineering what the best do naturally" is the name of the game here.
I hope this helps. They're both valid forms of lifting, just for different goal terms. StrongFirst for performance, Body by Science for bodybuilding and a bit of performance.
Also, it's interesting to note that both Arnold and Ronnie Coleman had very respectable powerlifting careers before they ever got into bodybuilding. Coleman did most of the work required for his physique by building his 800lb deadlift.
Fun to think about.