From a historical perspective, when did impact factors become so important?
I'll speak to my experience in the social sciences, and others are free to add their experiences from other disciplines. I see your question as part of a bigger question: more than scientometrics, when did quantification become prominent? And within what context? A recent paper published in The American Sociologist (https://link.springer.com/article/10.1007/s12108-017-9348-y) gives some insight to the question. The move towards quantification was gradual in America, from the 1960s onwards, when governments and institutions (i.e. Ford) began pouring more funding into quantitative research and essentially allowed it to become a privileged method of knowledge production. 

People started inspecting results, doing research, and even seeing reality in different ways, where everything was broken down into data. This was a big deal in the social sciences. We were beginning to measure what had been unmeasurable. New subdisciplines were springing up. But we needed a way to order and evaluate these rapidly growing disciplines. Impact factors entered into this picture down the line. They complemented these changes by providing an effective way of quantifying and measuring output, and, by extension, prestige. 

By contrast, in places where this quantification didn't happen the same way (Europe), impact factors are not used or referred to widely. In Europe, prestige has traditionally been more known, than it is measured in a quantitative sense. Rather, impact factors surely still exist, but don't carry the same weight. Particular journals and publishers, for instance, carry old-school, sedimented prestige from having published notable scholars in the past, etc. Today, this picture is beginning to change, and for the same reasons it did in America in the past. "Big-data" and quantification has begun dominating more funding, attention, and research, so impact factors are also becoming a bigger deal.