Civil War: T’Challa learns that capital punishment is wrong

It's really simple. Zemo wasn't a threat to T'Challa's throne. Killmonger challenged T'Challa for the throne and had already defeated T'Challa in combat once so Killmonger was a huge threat to T'Challa's throne if he lived. That's why T'Challa couldn't let Killmonger live. T'Challa had to eliminate the biggest threat to his throne.