I think the reason modern Minecraft and classic Minecraft (Pre 1.9 IMO the Elytra is the dividing line) feels so different is that modern Minecraft is trying to feel like a real mostly feasible low-fantasy world while classic Minecraft felt notably more gamey. I think the main thing more than anything else is the generation.
Chunk errors for example. They'd happen practically every time you switched versions (As in updating old worlds) and generated new chunks, so if you had a long running world you'd end up seeing quite a few of these. Modern Minecraft just doesn't have them anymore. They've fixed a lot of the world generation code so that regardless of differences in updated versions, new chunks generate based on old ones. It makes the world feel a lot more cohesive and seamless, compared to how you used to be able to see the errors in the game's programming.
Same thing with the far lands (which let's be honest despite knowing about them like 99% of people have never actually seen in person in game). They're a weird break in the game's code which has long since been patched out. Since I get the feeling I'll just be repeating myself on; 'Old generation was imperfect and would lead to the play occasionally seeing weird broken generation that had a strange vibe to it.' I'll just wrap it up with a mention of the true OG; Floating islands.
I guess to tie it together with a conclusion; The game has ultimately shifted focus overtime. Both Modern and Classic have distinct feeling to them. The main difference is in the generation code being sorted out overtime and more features being added to flesh out the world. Classic Minecraft is essentially a 'complete' sandbox in that the world is mostly empty aside from the landscape and occasional structure, leaving the player to be the only real deciding factor in marking the landscape. Modern Minecraft has far more features and biomes to make the world feel 'alive' in a way that gives the player more reasons to explore and build in different locations as well as question what the story of their world could've been.













