Because the government goes through a procedure for:

Finding the guy guilty without a doubt, and making this public.
Deciding on an appropriate punishment.
Exacting this punishment quickly and humanely.

You have to let the law do what it's there for. It's different than you killing the guy, because of the processes involved.

"I'm gonna kill that guy" is different from

"A crime has been committed."
"Is there evidence linking anyone?"
"Yes, there is.. Bob Bobson."
"Can we PROVE he did it?"
"Yes, we can.. he has been found guilty."
"Release the verdict to the public. What's the appropriate punishment?"
"Well, according to the law, and as everyone, including he, knows, it's the death penalty."
"Does he deserve it?"
"According to 7 judges, yes."
"Then he shall be put to death."