For about 50 years, that was what American Protestantism looked like–“mainline” denominations with fairly liberal theology and a strong emphasis on social justice, and a confusing network of more conservative denominations, independent churches, Bible colleges, radio stations, etc., which were generally considered less important and looked down on by the media and sociey as a whole. If reporters wanted a Protestant perspective on something, they were likely to go to a “mainline” theologian, not a "fundamentalist.
However, there were important changes going on in this period. On the one hand, many “mainline” theologians became unhappy with modernism and embraced something called “neo-orthodoxy,” which kept modernism’s respect for Biblical criticism and so forth but insisted that Christianity really was a revelation from God with doctrinal content and a message of judgment and redemption rather than simply humankind’s highest aspirations or something like that.
On the other hand–and here I finally start answering your question (!)–in the 1940s and 50s many of the more conservative theologians became dissatisfied with the “fundamentalist” label. This had come to mean not just “someone who believes in certain fundamental doctrines” but “someone who won’t associate with anyone who doesn’t believe exactly the same things.” Fundamentalists had become more and more quarrelsome and legalistic. So a new generation of leaders began calling themselves “evangelicals” as opposed to fundamentalists (fundamentalists, of course, would still insist that they are the real “evangelicals”). It’s a broader, gentler term, indicating that you believe in the basic doctrines of the faith but that your primary emphasis is on a personal relationship with Christ through saving faith. Billy Graham is the quintessential evangelical.
Very quickly, it became clear that even among those who weren’t hardline fundamentalists, there were important differences on how far one could go in this new openness. Many evangelicals said, and some continue to say, that their only real difference with fundamentalism is in their view of separation (i.e., they don’t separate from other Christians as readily or as strictly as fundamentalists). They still believe in the fundamentals, including a strict view of Biblical inerrancy and substitutionary atonement (the two that have come under most fire), and also including the belief that anyone who has not explicitly become a Christian in this life will go to hell.
Other “neo-evangelicals” became more open to theological currents such as neo-orthodoxy, which they saw as a move of God within the more liberal “mainline” churches. Over time, they became interested in Biblical criticism and found it harder and harder to believe in the conservative version of Biblical inerrancy. They studied the history of the doctrine of the atonement and weren’t sure that “penal substitution” (God’s justice demands punishment, so Jesus was punished in our place) is the only way to talk about it. (They’re also more likely to believe in women’s ordination.) To more conservative Evangelicals (who are more likely to capitalize the word), this is simply the same old liberal story. (In return, more liberal evangelicals tend to call these conservative Evangelicals “fundamentalists.”