Saws cut in one direction only
Tooth geometry, and why push saws and pull saws are built differently.
A saw's teeth are ground and angled to bite into the material on only one stroke, either the push or the pull, and simply skate over the surface doing almost nothing useful on the opposite stroke, because each tooth acts like a tiny angled chisel that only engages the material when it approaches from the direction it was shaped to cut on.
What is actually happening
Look closely at a saw's teeth and each one is not a simple symmetrical point but a small blade with a distinct leading face and a distinct trailing face, angled so that one direction of travel drives that leading face directly into the material at an angle steep enough to shave off a sliver of it, while the opposite direction of travel presents mostly the trailing face, sliding smoothly across the surface with little more than light friction. A traditional Western saw is built to cut on the push stroke, with its teeth angled forward, so pushing the blade drives each tooth's leading edge into the wood, while pulling it back mostly just repositions the blade for the next push. A Japanese-style pull saw reverses this arrangement entirely, with its teeth angled to bite on the pull stroke instead, and this is not simply a stylistic choice, since a blade cutting on the pull stroke is under tension while it works, being pulled straight rather than pushed, and a blade in tension cannot buckle or bow sideways the way a pushed blade can, which lets a pull saw be made noticeably thinner than an equivalent push saw for the same length without any risk of it flexing off course mid-cut. A push saw has no such option, since a thin blade pushed hard against resistant material bows sideways well before it cuts effectively, so its blade has to be made thicker and stiffer purely to survive being pushed at all, quite apart from whatever thickness the cutting itself would otherwise demand.
The cheese-grater comparison
Dragging a block of cheese across a box grater in one direction shreds it efficiently, catching against the small, angled, punched-metal teeth and peeling off in thin curls, while dragging the same block back across the grater the other way does almost nothing, sliding over the same teeth with barely any resistance because they are angled to catch the cheese moving one way and to let it slip past moving the other. Nobody grating cheese needs to think about this, the hand simply finds the direction that works because the other direction so obviously does not, and a saw's teeth are shaped for the same one-way bite for the same underlying reason, an angled tooth engaging material from one approach direction and sliding harmlessly past it from the other.
The number that matters here
Because a pull saw's blade is under tension while cutting rather than being pushed and therefore never needs the extra stiffness a push-stroke blade requires to resist buckling, its blade can often be made a fraction of the thickness of an equivalent push saw, leaving a narrower kerf that removes noticeably less material, and therefore less effort, with every cut.
Where this stops being true
Trying to force a saw to cut on its non-working stroke, pulling a push saw hard or pushing a pull saw hard, mostly just drags the trailing faces of the teeth across the material, dulling the edges through friction without actually removing any useful amount of wood, and wastes effort that a correctly used stroke would have converted into cutting. Recognising which stroke a given saw is built for, and using only that stroke to apply real cutting pressure while letting the return stroke simply reposition the blade, is what lets a saw cut cleanly and last far longer between sharpenings than a saw pushed and pulled with equal force in both directions. It also explains why a saw dragged backward through a cut it has only just started tends to catch and judder rather than glide, since the teeth that were meant only to reposition the blade are instead being asked to bite in reverse, against the very angle they were ground to avoid.