The simple answer is that expansions are cumulative and the game evolves with each expansion.
When you start a new character, you’re not playing WoW as it was in Vanilla (unless you’re playing Classic), you’re playing WoW as it is in BfA. The majority of the game systems (things like talent trees, how abilities are gained, and leveling in general) have changed significantly since that time. The way those aspects of the game are are consistent for all players, regardless of where they are from a progression and leveling standpoint.
New NPCs and “baseline” story are added with each expansion, but apart from basically being the setting and flavor of the expansion they’re in, they don’t really have any bearing on anything. Story is… not what I would call a strong point of WoW. I’ve been playing this game for almost 10 years at this point, and I honestly have not seen much logical progression for what story there is. There are tangential linking events between each expansions (although there’s a significant gap between the pre-Cataclysm and post-Cataclysm stories - the changes wrought by the Cataclysm eliminated any obvious links between early expansions, and MoP starts because some Horde and Alliance ships got lost at sea and shipwrecked, maybe because of changes brought about by the Cataclysm? That’s not really clear), but aside from one major event which occurs during “pre-patch”, which is removed following the release of the new expansion, there is no real flow of events.
Even within expansions, the story is fairly threadbare, with quests occasionally remembering that there’s supposed to be a story, but mostly remaining self-contained within each zone or quest hub if a story exists at all.
Oh, also one more point:
Technically talent trees were changed in MoP, not Cataclysm. Specifically they were changed to the current template in the MoP prepatch. Class trainers were likewise made completely obsolete at this time (they still offered skill training at the time I started playing about halfway into Cataclysm, although it was already restricted to only skills for your selected specialty).
Also, MoP added the Monk class, and introduced the precursor to the current Mythic+ with challenge mode dungeons.
The profession system was also revamped for the first time in Legion, moving from a progressive system to one where you could start from any point, and adding tiered recipes with reduced material costs for higher ranks.
This was changed again in BfA to a stratified system where each expansion’s profession ranks leveled independently - so rather than professions going from 1-800, each expansion has its own tier from 1-150 (or whatever) completely independent of other tiers.