There would have been no Albus Severus Potter.

 

Actually I take back the no incest part of my post. I don't imagine that there would be some freaky stuff going on at the Burrow, but the books kind of gloss over how Purebloods keep their lines pure by regularly intermarrying and therefore are all closely related to each other. GRRM would probably not gloss that over and would play up the racism/elitism aspect of it. Arranged marriages, intentionally keeping out mudbloods and foreigners, there being a hierarchy among the purebloods based on how old and pure your bloodline is. Possibly an extension of how the Gaunts were a little inbred looking to some of the others--Crabbe and Goyle not looking brutish because they fit the role of thugs, but because of some genetic degradation.

It's not like England had some inbreeding problems among royalty in the past that he can draw from or anything.

 

I'd be down to read HP without "muh power of love"

 

I don't know. Then I couldn't live in a world where I can assume Huey Lewis is very confused as to why a bunch of Muggles enjoy his song about ancient magic.

 

Among other things, we would have much better villains.

 

I'd also imagine that there would be less plot armor for the sake of poetic justice. For example, Neville taking on Bellatrix at the end to avenge his parents only to get killed because he's a barely graduated teenager and she's an experienced terrorist witch. But then because plot armor doesn't apply to her either, she gets oneshotted in the back of the head by Molly Weasley.

 

I agree GRRM would bring a new dimension to Harry Potter. Imagine if Ron Weasley was actually killed when he sacrificed himself in that giant game of chess in the Sorcerer's Stone, Harry Potter was eaten by the basilisk in the Chamber of Secrets and Ginny is permanently possessed by the spirit of Voldemort. Hermione was horribly maimed by a maddened Remus Lupus in the Prisoner of Azkaban, and slowly becomes addicted to Polymorph juice that allows her to maintain her appearance. Leaving Neville Longbottom to take up the mantle of the boy of the prophecy, but he is seduced/manipulated by a disguised Beatrix Lestrange into believing that it was really Dumbledore who drove his parents mad, leading him to betray Dumbledore at the lake in the Crystal Cave by the sea and allow him to be pulled in by the Inferi there while he's under the influence of the Drink of Despair etc the possibilities are endless.

 

The books would be longer, more complex and have a darker tone even if they are still written for children. It wouldn't be ASOIAF, but Harry Potter would definitively die at the end of the story...oh what I am talking about? Whom I am kidding?
He would never finish it.

I am still a fan of GRRM, but based on what we have seen, we can conclude that as soon as Harry Potter became mainstream and popular and never-ending flow of cash ensued, GRRM would be distracted with other things and lack motivation.
At first he would write huge sequels that would make Order of The Phoenix look thin (i.e. too long) and introduce dozens of complex characters that weren't really necessary. Think of "A Dance with Dragons", that monstrosity dragged on for too long.
He would regularly check fan forums and change direction of the story if fans correctly guesses what's going to happen. I am still upset at what he did to Tyrion in book 5, what a huge waste of time his entire journey was, and all of it to keep the element of surprise.
After years have passed without GRRM writing a single Harry Potter book or announcing any date or even an approximate year, he would publish entire books about Harry Potter history and lore with no actual sequel in sight. He would suffer from writer's block when it comes to Harry Potter series. And then he would have ceased publishing anything noteworthy at all.
....writing this made me feel depressed. Now I need to read the most depraved fan fiction I can find.

I also have to add that what makes HP popular among fan fiction writters is simplicity of Harry Potter. In comparison both GOT TV series and ASOIAF have very few fan fiction.

 

Diary Tom would have raped Ginny. Barty Jr would have tortured Neville before wiping his memory of it in GoF. Molly would have stolen from Harry's vault. Lockhart would have slept his way through the 7th years. Snape would have a side business selling hairs/etc of the students/faculty. The Umbridge/Centaurs scene would have been explicit, and either witnessed by Hermione and co or included them. Someone would have been left behind at Malfoy Manor to be tortured, and their suffering revealed at the battle for Hogwarts. Jesus, the Hogwarts occupation would have been brutal. Everyone except purebloods who toed the line would be used and abused.

Basically what I'm saying is that it would have been shit. Although there'd likely be some very interesting world building done.