Given the fantastic nature of Anakin's birth the difference is academic. It's not like having the last name Skywalker made Shmi special.
Honestly this. Rey is Skywalker in name only and makes a huge difference. Skywalker post-Anakin is no longer a last name like Jones or Jinn. It’s a marker that indicates that the person is someone with the lineage of the one and thus possesses their unrivaled force potential.
It’s why the Skywalker name carries so much weight to people in the know. Anakin was the one and had unrivaled force potential because of it. Being of his lineage, Luke, Leia, and Ben also have that incredible potential. Han is technically also part of the “Skywalker family” but no one expects him to be a super Jedi because he doesn’t share the lineage.
Same with Rey. She does have a strong lineage (the Palpatines) but Sheev pales in comparison to Anakin in terms of potential (Anakin at full potential would be twice as strong as Sheev).
Disney f’ed up the trilogy by killing off Ben. It honestly would have been better if Rey died, ending the Palpatine line and having Ben go on a redemption quest, learning how the next line of Jedi should be as one who has experience in both the light and dark sides of the force. Rey wasn’t chosen to become a Skywalker, he has no relationship apart from being a student (and arguably learnt little based on the comics). Rey literally is Skywalker in name only.
A couple of thoughts.
First, I think the "End of the Skywalker Saga" bit is really just marketing fluff. For one thing, I'm old enough to remember when Friday the 13th Part IV was subtitled "The Final Chapter."
They made 7 more films in the series, before rebooting it with an 8th, and then dropping it (for now....it'll be back, I'm sure). The
penultimate one (Jason Goes to Hell) was also subtitled "The Final Friday." Then they made Jason In Space and Jason vs. Freddy.
Forget it, Jake. It's marketing.
Second, I think with the ST, "Skywalker" has become less bloodline and more a title. But let's be clear: JJ's rather myopic vision and storytelling chops (or lack thereof) are what lead us to the somewhat confusing outcome of Palpatine becoming an honorary Skywalker, and leaving things there.
The saga of the Skywalker bloodline is indeed over with Ben's death (and Leia's and Luke's). That doesn't mean that "Skywalker" couldn't become synonymous or even a neologism for "Jedi." Perhaps if we ever see a post-ST set of films, Rey will lead "The Skywalker Order" or something, and instead of being an institutional entity like the Jedi at the fall of the Republic, they'll be more like wandering do-gooders who battle those who would use the Force to harm or dominate others. Knights Errant, if you will, rather than highly organized, centralized and regimented force.
Anyway, I don't think it matters a ton one way or the other. Again, it's all just marketing nonsense. If they brought Palpatine back, there's no reason why they couldn't, I dunno, infuse Rey's midichlorians with Skywalker chloriquantofaboonty or whatever and now, bim-bam-boom, she's REALLY a Skywalker. Maybe when you join the Skywalkers, you give up your surname and become "Han Skywalker" or whatever.
We've all remarked that the Sequel Trilogy seemed to lack a real sense of coherent vision, and I do fault the producers and Abrams for that. (Johnson less so because they could've said "No, Rian, you aren't allowed to do that." Plus I like his ideas a lot more than JJ's.) It ends up striking me as being about (1) a focus on marketability, and (2) fan service primarily focused around recreating the OT. And beyond that, there wasn't much vision. Probably some hastily thrown-together ideas, but otherwise....yeah, more about marketing.
I'm betting that post Skywalker Saga is off limits until the films go there. They aren't going to be able to maintain the thin facade of 'its all canon' now if they let a bunch of writers loose on that section of timeline.
And that'd be very smart unless and until they decide to push the timeline past where those characters are (e.g., a couple generations into the future).