So, in older editions of the game, XP made a lot of sense. But this was because of a few features of those games:
1. Classes needed different amounts of XP to progress to the next level.
2. Characters with high ability scores earned more XP than those with low scores.
3. Each class had a few optional bonus ways to earn XP.
In addition, older games tended towards this idea that PCs didn't always adventure together - sometimes, three players would grab whatever PC they had on hand to go explore the DM's dungeon. It wasn't always the same five or six PCs adventuring together - the party composition was much more in flux.
With that in mind, it was pretty much a given that every PC had different amounts of XP, and there was a wider spread of levels in the adventuring party.
Compare that to modern D&D versions, and things are a bit different. Each class progresses at the same XP limits, there are no bonus methods to gain XP in the core game, no XP bonuses for ability scores, etc. Plus, the party is assumed to be static - it is, I'd guess, more common than not for every PC around the table to have the exact same XP total (or near enough).
I'd say that if every player at the table has the same XP amount written on his or her character sheet, experience points are really just an illusion. By having the DM say "at the end of this adventure, you all level" or something similar, it saves a bit of bookkeeping and really doesn't change much.
Plus, as it was said earlier, players do what they're incentivized to do. If you remove XP, you are incentivizing the players in a different manner. If you level PCs every six weeks of real play, for example, the characters can really do whatever the players most enjoy. If you level them up when they progress through the adventure, you're rewarding players for keeping the game moving.
It creates a different thing than "Man, I need 50 more XP. Who wants to go orc hunting?"
I've mostly done the XP thing in the past. Lately, I've made a tentative switch to milestone and found my group loves it. In my next campaign, starting next month, we'll be following a "1 level per adventure" format that I think will be fun.