I have searched, this on AH.com as well, here are the answers, and also according to official documents:
1.) Having Khomeini killed will make the Islamist turn to theocracy impossible. It was because of Khomeini's hijacking that Iran turned Islamist.
(Do you know Khomeini was an obscure figure by the 1970s, that only an article in January 1978 made him known again?) Butterfly this, and you will not have the immediate cause of the protests.
2.) Having the status quo in place, so you won't have much protests for the late 1970s. But by 1980, the Shah will die. So his death sparks protests in Iran, but because the Islamists don't have a rallying figure, they don't take power in IOTL. Then, the new Shah, Reza Pahlavi, along with Empress Farah, acquiesces to the demands of the liberal opposition. He then pulls back the repressive acts of his father, and scales down some reforms that anger the Shia clergy and many in the muslim population.
3.) Having a pro-US Iran will butterfly away the Iran-Iraq war (no Khomeini purges), so much more peaceful Middle East).
4.) The Soviets will invade Afghanistan anyway, but it depends on the circumstances. [If Taraki (the moderate in the PDPA) manages to negotiate with the Muslim population of Afghanistan to scale down the reforms, the most extreme ones, we might have the Soviets win there in 1980, so more stable USSR, but Iran may or may not intervene]. Not sure what will happen here though. But since oil prices will not go through the roof, the Soviets will have less money, so maybe more sweeping economic reforms in the early 1980s. It could go either way.
5.) Political Islam will certainly be much less influential here.
6.) So in the long run, Iran turns to a democracy.