Iron woody has no reason to lie now do they? Could it be they are trying to sell you a cheaper product and pass it off as the same quality, companies have been known to do that, now haven't they?
I can tell you from experience because I have used both, multiple times, they are not the same. I used woody bands twice on a deadlift platform and they popped within 3 pulls off of it, not wear and tear put popped.
Thank God it wasn't during warmups and not during a heavy attempt. Been using the JS for many sessions ever since, yet to have one pop, but they will wear over time. If you aren't pulling with them then, it doesn't matter, the Woody's work fine.
Just because a manufacturer, whose job it is to sell you their product, tells you they are the same, doesn't mean they are the same. I don't have a dog in this fight or am I sponsored by either or anything like that, I have no reason to steer you wrong, but I will speak up when I feel something could be unsafe and like I said, if you are planning on pulling heavy deads with bands, then go with the JS.