FMJ bullets have a Full Metal Jacket covering the tip of the which prevents them from expanding, ie, military bullets. These are the bullets that many states prohibit for hunting big game animals. However large caliber solid bullets are recommended for hunting very large animals, like elephants, where you want maximum penetration.
Mono copper bullets, like Barnes X,some Hornadys, etc, have a very small hollow point and a hollow cavity in the bullet tip or a plastic tip which almost always insures expansion, and are very effective on game animals.
FMJ are harder metal alloy covering softer core. Solid copper is not FMJ. Copper is entirely a softer metal than the stuff used to jacket FMJ bullets.
FMJ bullets were primarily designed for military use after various wartime conventons banned expanding bullets.
The Full Metal Jackets are copper or a copper alloy over a lead or steel core for amor piercing bullets. The copper jackets engage the rifling of the barrel and prevent leading of the barrel or damage to the barrel from steel core bullets. The copper jackets are alloyed to harden the copper to help prevent copper fouling in the barrel.
Although some bullet manufacturers may "tweak" the alloys of the copper jackets to make their bullets "better" than the other guy's bullets, the jackets are still a copper alloy and basically the same copper alloy is used in military FMJ bullets as is the copper alloy used in the jackets of standard lead core hunting bullets.
Mono copper bullets are also alloyed to help prevent copper fouling in the barrels and these bullets have multiple grouves in their sides to reduce the amount of copper to steel contact as the bullet moves down the barrel.