What happens instead?
I doubt John XXIII would have allowed birth control, BTW. But the real question is, will we see a different decision, if so, what kind of decision (e.g. the Vatican make distinctions between types of contraception or not), or just no decision (most of the time popes try not to touch difficult issues unless they have to).
I doubt it would make a big difference in sexual mores. The fact is that millions of Catholics today love the Church and refuse to listen to a word it says on sex. The transition would be easier, with a couiple (million) fewer unhappy marriages, ruined healthy and guilt-racked young people, but in the big picture of history that doesn't tend to matter.
It would make the campaign against AIDS much, much, much easier. These things are hard to quantify, but a few hundred thousand fewer deaths in Africa and Latin America are not too improbable without people being told that the virus slips through nonexistent pores in condoms or that they're going to hell if they listen to health workers.
The biggest change could well come in terms of the political and social influence of the Catholic Church. With Humanae Vitae, the unerring Catholic instinct to identify the tide of history and staunchly oppose it once again came to the fore. At a time when, conditioned by the horrors of WWII and the high tide of Christian democracy (before it became a trademark only), its credibility was at an all-time high, Paul gave the church its Prohibition moment. From Humanae Vitae onwards, people all over the Western world (and soon enough the developing world) would associate Catholic teaching with 'banning the pill' and assume it could be safely disregarded (at every Catholic youth gathering today, the waste includes thousands of used condoms - which is rather like finding empty whisky bottles in the wastebasket after Milli Görüs retreats).
Imagine a world where the Catholic Church retains a concrete political authority among the faithful similar to that enjoyed by the Moral Majority. A world where Conservative politicians didn't enjoy the luxury of assuming that whenever the pope disagreed with them, they could safely ignore it because everyone does. A world where Catholic criticism of capitalist economics *matters* to the minds of millions.
Things Get Interesting.