Compound = pulley bow (modern)
Composite = multiple materials
Self bow = one material.
Composite material bows are more effective per length. . .and semi-composite bows like the English longbow (which used staves drawn from the transition line between sapwood and heartwood to get the composite effect of having different tensile strengths on front and back) us variations in the tensile character of the face and back of the bow to get more power out of less bow material.
Further complexities, like bows built of 3, 5 or 7 integrated parts of differing materials, like different woods, horn, or metal allows you to mix and match material types to get better characteristics out of given portions of the bow, like you want the central handle to flex as little as possible, so it doesn't push the arrow in any direction other than forward, you want more flex and strength out of the lengths above and below the handle, and then there are characteristics you'd want to add to the tips both in terms of durability, string wear, and consistent release without vibration or a curved surge of power transfer.
The pulleys and such of modern bows merely allow you to pull and hold with ease. . .the pull is just as hard, but once pulled, the mechanical advantage makes it easier to hold. . .and it's much easier to aim a 50+ pound draw bow if you're not holding 50+ pounds of tension at bay. . .so the reduction in effort jitter and ability to walk around holding the bow pulled are the advantage to those.
Self bows, like those of many cultures, are relatively weak compared to their composite cousins. . .the bows of the native americans for instance are nothing close to the mongol bow. . .and even the semi-composite english longbow is merely a half step in the right direction. . .the composite bows, especially late era ones like the mongol bow, were a step forward akin to smoothbores vs rifles in firearms.
As to arrow material, other than the minor issues noted above, I've never heard much that points to vast differences, compared to bow materials, which per above, matter quite a bit.