in 5E, the assumed story is allowed to influence the combat rules in ways that it wasn't in 4E.
I'm not sure I follow this.
Two examples from 4e:
Come and Get It: the goons charge the Conan-esque (or Jet Li-esque) fighter, who cuts them down as they come adjacent. That looks like a combat rule that has been influenced by the assumed story.
Valiant Smite: the paladin becomes more likely to hit the more foes surround him/her. That also looks like a combat rule that has been influenced by the assumed story.
If an instance of "assumed story" in 5e is Fireball as a signature spell, I'm not sure how 4e differs - classes had signature moves, and signature orientations more generally (in virtue of "role" mechanics, preferred stats, etc).
5e's design focus seems to be more on the character's role in the game 'world' over what he can DO in said world.
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As you proceed through character creation in 5e, there are a lot of questions asked to help a new player create a persona. When they pick a race or class, there are suggestions as to why they became a Ranger, or where they got their Wizard training.
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In my experience, 4e gives you a paragraph about your class
Again, this doesn't fit with my experience with 4e - each class entry in the PHB, for instance, has an outline of the class's typical outlook(s), a discussion of how members of that class relate to various gods and are found among various races, etc. And each race entry has 3 vignettes illustrating how a member of that race might become an adventurer of some class or other, with some goal/focus or other.
I'm not sure exactly what distinction you are drawing between a PC's role and what a PC can do - "role" is a word that overlaps in meaning with words like "office", "function", "position" etc, which are all about what a person does, or is permitted/expected to do. But anyway, the 5e classes seem to me to be set out in ways quite similar to 4e ones - proficiency lists (for amour, weapons and skills), save proficiencies (replacing defence bonuses), and then lists of abilities by level.
What is innovative in 5e (from the D&D perspective - not from the perspective of RPGing more genrally) is the personality/inspiration system.