Thinking back, the first game that seemed to me to move beyond the classic gameplay formulas was Sonic.
As a platform game, it felt so different to previous platform games. It was fast and lively, and full of strongly animated graphics. It set a new pace, and that pace isn't so different to the pace we still have today. One might even describe the pace of modern games as being hyperactive.
The controls became more fluid and floaty, whereas in previous games they were pixel perfect stepping movements. This is something else we seem to have retained. We no longer nudge our game characters / things in quantum steps; instead we accelerate them.
The structure of the levels in Sonic was less apparent as the graphics fused together much more coherently, whereas before the structure of the platforms in older games were obvious. Sonic gave us graphics with a more natural appearance, but less easy to read. Less obvious where something ended and something else began. The new graphics was less abstract. In many ways, that has continued in modern games.
Throughout the 1990s, these changes gradually became the norm in computer game design, and are pretty much ubiquitous now, and even in retro remakes the developers can't avoid the temptation of using some of these new features.
But all of these elements affect gameplay, sometimes negatively. And I think what the continued attraction is to old school games is in the absence of these elements, because games can be a lot of fun without them.