Why should we Americans care whether or not a murderer is considered insane in the eyes of the court?

Do those considered mentally insane receive something different than those who are considered mentally stable but both are murderers?

I don’t understand the difference… Does the mentally insane get off the hook with the death sentence? And why would those defense lawyers want to work with someone who is a mass murderer?

I’m not in favor of the death penalty because it teaches that killing another is a way to solve a problem. But someone who is mentally insane and a murderer deserves something close to the penalty.

Do they do intensive research at the insane asylum or what?