SCOTUS did not rule in Furman v. Georgia that capital punishment was cruel and unusual under the Eighth Amendment. They were concerned with the way the death penalty was being used for sentencing, particularly a perceived bias for certain groups (like black offenders). The ruling was highly contentious, there was no consensus within the Court, and no Justice joined the opinion of another, which was unusual. It's also why it was such a short decision, only one page. Only Justices Brennan and Marshall believed that the death penalty should not be used in any instance. The other Justices that made up the majority were more concerned with vague statutes and sentencing procedures that resulted in cruel and unusual punishment. In particular, they took issue with statutes that mandated the death sentence.

Gregg v. Georgia ended the temporary (and unofficial) ban on the death penalty as legislatures tried to more clearly define capital punishment statutes. It laid out the criteria lawmakers must meet if they wanted their sentences upheld.

It might be important to note that Douglas was no longer on the Court and replaced by Stevens, but it didn't really have an effect on the ruling.

Relevant quote from one of those who did in 1976:

"From this day forward, I no longer shall tinker with the machinery of death. For more than 20 years I have endeavored...to develop...rules that would lend more than the mere appearance of fairness to the death penalty endeavor...Rather than continue to coddle the court's delusion that the desired level of fairness has been achieved...I feel...obligated simply to concede that the death penalty experiment has failed. It is virtually self-evident to me now that no combination of procedural rules or substantive regulations ever can save the death penalty from its inherent constitutional deficiencies... Perhaps one day this court will develop procedural rules or verbal formulas that actually will provide consistency, fairness and reliability in a capital-sentencing scheme. I am not optimistic that such a day will come. I am more optimistic, though, that this court eventually will conclude that the effort to eliminate arbitrariness while preserving fairness 'in the infliction of [death] is so plainly doomed to failure that it and the death penalty must be abandoned altogether.' (Godfrey v. Georgia, 1980) I may not live to see that day, but I have faith that eventually it will arrive. The path the court has chosen lessen us all."

Supreme Court Justice Harry A. Blackmun, from an opinion dissenting from the Supreme Court's decision denying review in a Texas death penalty case, Callins v. Collins, Feb. 22, 1994.

In my opinion, the best source on this topic is Bob Woodward and Scott Armstrong's The Brethren, which examined this period of the Supreme Court's history and spoke in-depth about the death penalty decisions. The book contains a lot of insider information from law clerks from the period, and it's behind-the-scenes look is interesting (and controversial).