###In short: Symbolism
I think Tolkien was using the forging of Anduril to amplify the symbolic effect of the Return of the King as a counter point to the rise of Sauron. There is a good bit of reincarnation and subsequent renaming throughout the trilogy, and this is one instance.
It's also worth noting that "Narsil", in Quenya, means "white and red flame", symbolizing the Sun and Moon, or "the enemies of darkness". Andúril means "Flame of the West", meant to more directly oppose the darkness to the East in Mordor.
In response to @NKCampbell's comment, I disagree (in this context). Whether or not the shards of Narsil were forge welded back together or were smelted into a new blank, Anduril is a new sword. As @Werrf says, this is a philosophical question, but it does differ slightly from Theseus's Paradox, as the same steel was used, whereas in the paradox, the ship was eventually made of entirely different material.
Very bright was that sword when it was made whole again
On that, we agree. I don't think it was smelted down. But I still say it's a new sword.
The question here is, "How much does a sword need to change to become a different sword?"
First, it went through a process. It was reforged, and technically, the molecular structure was changed by the welding, hardening, and heat treating of the steel. Welded steel is strongest at the joints. The sword would feel different in use. It would flex differently. It would sound different on striking. This would be also be true if The Shards were treated as revert and the smelting process were performed again. Also, it was engraved, almost ceremoniously. And then it was renamed. As I said, this is a significant thing in Middle Earth.
Second, it can be debated that when a Thing breaks, it's no longer a Thing; it's now a Different Thing.
- When you take a
plankand run it through a chipper, it's nowmulch. Same physical material, but not as good for making furniture. - Is it really a TV if you can't turn it on, i.e. perform the most basic function of a TV? I call this garbage (until I fix it).
- Better still: if a plank in a table splits, and you mend it with some butterfly inlays, is it the same plank? Seems to me it's now two planks mended together...
- Another place to direct this thought experiment is to the Japanese practice of Kintsugi, in which a broken piece of pottery is mended with gold. It becomes both stronger and more valuable than the original.
Third, and perhaps most important in a literary sense, Narsil was a named weapon. Named weapons are a common trope in the Fantasy genre, and are almost characters. I'd argue that in this context, Narsil has died and reincarnated as Anduril, imbued with a new spirit. It exists for a new purpose (or rather, the same purpose but for a different person). See Gandalf the Grey => Gandalf the White.