Copper hunting bullets can have some advantages, some disadvantages too. Compared to lead bullets they retain 98%-100% of their weight after impact compared to 50%-80% for jacketed lead bullets. This means a quite light copper bullet will match the penetration of much heavier lead bullets.
Disadvantages... They have to impact at much higher speeds to ensure expansion. Shoot them too slow and they often do not expand and act like a FMJ. A common mistake is to shoot the same weights in copper as you'd use with lead. This often leads to slow impact speeds and poor results. For best results drop down at least 1 bullet weight, maybe 2, than you'd normally use with lead bullets.
I'm loading 130 gr Barnes in my 308 @ 3050 fps. That is about the same speed as 180's from a 300 magnum. At closer ranges the 2 will impact at about the same speed,. The 130 copper bullet will end up weighing the same or more than the 180 gr lead bullet fired from a 300 mag and give about the same results.
But at mid to long range the 130 gr bullet will quickly slow down and be too slow to expand. The heavier 180 gr bullet from the 300 mag will retain energy and speed much longer and it needs less speed to expand so it is the better choice at longer ranges.
Used properly they can make less powerful guns suitable for bigger than normal game. At least as long as the range is kept reasonable.
I'm not as up to date on copper handgun bullets, but would think that the slower handgun speeds would mean less, if any improvement over lead.