So, here's a description of the problem as it has been experienced by me at least:
In some hypothetical world 6e uses the "fiends are just aliens" stuff, and a new player stoked to play D&D comes to your table all excited about playing her character who is a bard whose heroic origin includes a story wherein she made peace between a faction of demons and her hometown, leading to a mutual prospering where the demons even defend her hometown from attack by other creatures. It's part of her identity as a persuasive, diplomatic character - she managed to make peace with demons. Truly, an exceptional and heroic feat! It's why she's taken a "peaceful diplomat" bard build in 6e and why her character has ranks in Religion - to let her know what the demons want. Maybe she's tiefling, too - with the pre-4e "varied children of the planes" story, and that plays into her character's success there.
Of course, in your campaign, this is impossible - demons are not creatures you can make peace with.
The price you then pay is in telling your excited newbie that she can't play the character she's really excited about playing. Sorry, I use old lore, demons aren't like that in my games. You can be a heroic peaceful diplomat, but maybe you made peace with some orcs instead? Maybe instead of a tiefling...you're a half-orc?
Or, maybe she doesn't even tell you the backstory. You know, she has no reason to expect that any other backstory is necessary, 6e presents this narrative as THE narrative. And so when she tries to make peace with the rampaging demons, suddenly everyone else at the table is like "what are you doing you fool?" and she's like "oh, this is a situation wherein everyone tells the hero that she can't do it, but I'm going to be the hero and go forth and do it!" and then she gets clawed to death instantly (or, you have that same conversation above and nuke her character concept, but now it's a few sessions in because the DM and the player weren't on the same page about the lore).
Or maybe your game is ambivalent about demons and you don't think your campaign will really use them, so you say, "sure, that character's fine," and then some other player who's been by your side since 4e sees this bard bragging about making peace with demons and now in his cleric's mind this bard is a liar and a fool, dangerously naive and possibly inventing memories when she talks about the demons who defend her hometown.
All of these situations are part of the cost of changing lore. That cost might be felt differently by different players using and caring about different lore. That cost can be worth it or not, and can be mitigated or intensified in various ways. It's not smart to pretend that this cost is non-existent or only comes from some ulterior motive related to the lore you favor, though.