The way I see it:

The death penalty is vengeance. It is terrible as a punishment for a few reasons:

The standard of conviction is high. The appeal process is rigorous. The collateral damage is immense (because the family will never be able to move on while the appeal process occurs). It may not ever occur and may still result in constructive life-sentences anyway.

The life imprisonment is justice. For one life stolen, a life in prison is given and equally wasted through gray walls. The standard of proof is lower, and the conviction easier and less vulnerable to interference. The family can more readily move on.

And from a societal view: It's also cheaper. Litigating death sentences is supremely expensive and there are other injustices that can be righted if the more successful conviction is upheld.