Right off the bat we have the change in terms from “Inspiration” (PHB14, p. 125) to “Heroic Inspiration” (PHB24, p.368), but there’s more difference than simple vocabulary.
The 2014 Inspiration was limited to d20 rolls - specifically: attack, ability, or saving throws - and imparted Advantage on the roll. It could be awarded by the DM for playing your character’s personality, and other players could award their Inspiration to you as well.
In 2024, Heroic Inspiration is still awarded by the DM, but can also be earned by particular game mechanics (human characters gain it every dawn). Players are not able to give up their Heroic Inspiration to each other, but a player who already possesses it may, on receiving another, pass it instead to someone without it. The big difference here is that it can be used to replace any die roll, not just a d20, with the trade-off that the newly-rolled value must be used.
This next bit is a little disappointing, but the DMG2024 barely mentions Heroic Inspiration, burying it in a “look at the PHB” sidebar on page 46. This is a sharp contrast to the DMG14 giving a two-page spread (pp. 240-241) on different considerations of genre, player engagement, and group dynamics.
This particular change feels like WotC is trying to formalize and codify what was once an open-ended, vibes-based DM tool.
In my own experience from '14, Inspiration was an award that I had to get creative with finding ways to coax players into using - physical tokens did the trick nicely - and I admit to being guilty of not having read the entirety of any of the source books prior to playing or DMing.
I actually preferred the '14 Inspiration awarding Advantage to '24’s more restrictive (and possibly punishing) result replacement, even if the '24 version works on a greater variety of rolls. Looking across the landscape of games, there are more open-ended mechanics and features that I would greatly prefer to what we’ve gotten with this update to Heroic Inspiration.