It makes a difference primarily because he abandons his blue saber in becoming Darth Vader. It doesn't become a tool of evil as in the original cut.
Giving "Darth Anakin" a special saber for him as a sith means the saber Luke gets is still the weapon of a good man and hasn't been used to murder children.
Obi Wan saying "your father wanted you to have this" is still a bit of a lie, but less so. i.e. he WOULD have wanted you to have this, had he not turned to the dark side. It emphasises that Anakin "died" when he renounced his saber. Luke's father is not the same person that he defeats on Mustafar.