Hi ZT. I agree that there are substantial obvious differences between butterfly and freestyle. The less obvious things are the substantial similarities.
Yesterday I did some experiments with body dolphin (no arm stroke) in two different positions, first with my arms at my sides and hands at my hips, and second with my arms extended. The undulation is significantly easier with arms extended, which means that (at least in my case) I am using my arms somewhat to facilitate my undulation. That said, the basic principles of energy conservation still hold, and you are working harder without going faster if you aren't pushing water backwards. It may look like the swimmer is pressing water down to get out of the water, but I claim that isn't true. For me, one of the hard things to learn when doing fly is to avoid doing exactly that. Instead you should use a patient catch. This means that you try to get your arms in the position to push water back by moving them into this position while exerting the minimal pressure possible on them, just like in freestyle. I might add that I have no trouble at all getting my head out of the water to breath when I do body dolphin with my arms extended overhead but no stroke, so you really don't need much arm work to do this. The main motion is guided by your hip undulation. The temptation for beginners is also to lift the head out of the water (just like in freestyle) and this also is unnecessary. When your undulation is working properly, the head comes out in an effortless fashion without breaking spinal alignment.
My guiding principle these days when swimming fly is to try to do as much as possible with my body and as little as possible with my arms. This means a lot less stress on my old broken shoulders.