I am comparing the 1946 FA Cup final highlights with that from just 30 years later, in 1976.
They have completely different atmospheres. Some of these differences are technical advances which were available by 1976: colour, slow motion, and being able to have multiple cameras cover the action.
But there's a lot more.
In 1946, the commentator was calm, collected, and almost an objective observer. By 1976, the commentator sounds like a fan himself: excited by the action.
The way the footage was cut and commented on may explain a lot of the differences. But there is still yet more.
The crowd celebrate goals in 1946 by, it seems, mainly waving their hats. I see the odd rattle. And lots of people saying "jolly good show" (or words to that effect). By 1976, however (as seen around 3:07 in the video), the Southampton bench are running onto the pitch, the crowd are jumping around wildly, and the players themselves celebrate with hugs and jumping on each other, rather than the rather formal handshakes in 1946.
Compare the footage at around 02:00 in the 1946 video: that is an extra-time goal which puts his team into the lead! But judging the reactions of his fellow players by the standards of 1976, you'd think that the scorer had just returned from a short holiday and was being met by his family at the airport: "hello, old chap - good trip? Good. Your mother is bringing the car around."
This is the same sport. Actually, the very same sporting event in the calendar. It's the same venue. Just 30 years separate these events: so why are they so very different in atmosphere?
For the tone of the commentators, one critical difference is that the 1976 broadcast was live commentary, but the 1946 one shown was a Pathé newsreel - the commentary was dubbed later.
I can't comment on whether live commentary in 1946 (e.g. the radio) would be less exuberant than 1976 and whether this trend (if any) was mirrored by crowd behaviour without entering into speculation. Social behaviour (post Antiquity at any rate) is well outside my area of modest understanding.
Have you tried also posting this on r/soccer? Quite a few football enthusiasts who might know the answer.
It is correct to say that this is bound up in a lot of the social change that was going on in post-war Britain but I think its worth looking at some of the specific transitions within English football to really answer your question. Within all that change I think you can genuinely point to one very significant figure, Jimmy Hill.
First lets look at the actual style of broadcasting. Even when you compare the '76 commentary with the commentary by Kenneth Wolstenhome for the '66 world cup final there are marked differences. Wolstenhome has a public school accent with BBC pronunciation and David Coleman had a northern twang and spoke 'normally'. This step change in the nature of football commentary came about within the BBC only after the appointment of Jimmy Hill to London Weekend Television in the late 60's and specifically how he shaped their coverage of the 1970 world cup. LWT was part of ITV which was established in 1968 and fits in with the wider social change that I mentioned earlier. The formal institutional, public school boy run BBC no longer had the monopoly on football coverage and LWT set out to innovate.
Prior to this world cup, football coverage was very formal and newslike and Jimmy Hill and LWT turned this on its head by employing an expert panel that debated heavily and was often controversial. This discussion between Mullary and Allison is a good example of how more edgey it was than the bland bbc coverage. [This is not from the world cup but is LWT and they are actually following up on a bitch fest from the world cup coverage in which Mullary was heavily criticised]. They also used Brian Moore as a commentator who they had picked up from the BBC and effectively let him have free reign and allowed him to effectively shout into the mic.
David Coleman (the commentator in '76) and his style were the BBC's response to LWT's innovations and the commentary of Brian Moore. In just a few short years, football coverage was blown wide open in terms of format and style and this is largely because of independent television and specifically Jimmy Hills innovations.
The second aspect that you picked up on was the attitude and reactions of the players. Once again this was part of a wider social change but again Jimmy Hill was instrumental. During the late 50's Jimmy Hill was the chairman of the Professional Footballers Association and was instrumental in having the wage cap lifted. AT the time of the 1946 final the maximum wage was £12. Wage caps continued to keep wages at exceptionally modest levels of around £20 per week which was only about 25% above average industrial wages.
Once this wage cap was lifted in 1961, wages grew exponentially so that just one year after the 1976 cup final, Kevin Keegan, a promising young player about to sign for Liverpool who were then the biggest club in the land, was on approximately £130 per week. That's more than 10 times the wages of 30 years previous. In 1976 the average weekly wage was around £32 so we can see that the ratio had shifted quite a bit. This was player power and with it players became stars who were far more vocal on the pitch, full professional athletes and placed less emphasis on amateur ethics that were still prevalent in post-war football.
Jimmy Hill was at the center of both these changes and was a central figure in the huge shift in the national game.
Sources: Taylor, M. (2013). The association game: a history of British football. Routledge. [Disclaimer: this book does not support the narrative that this is all attributable to Hill but does a better job at showing how this evolution took place in a broader social setting and over a longer time period]