I believe that Narsil is more important to Tolkien's overall story, with its themes of redemption, perseverance, rebirth, etc. It is the most symbolic of blades, its breaking symbolizing the sundering of Men from the rest of Arda, and also Isildur's corruption. Yet, like all things that come from good, it was reforged, remade and renamed so that a new King could fulfill his part of the "covenant", so to speak. In Narsil Tolkien is telling us that regardless of our flaws and shortcomings as mortal beings, we still have the capacity to do great deeds. Narsil is Man's heritage, and perfectly illustrates both our strengths and weaknesses: yes it was able to cut the Ring from Sauron's finger, but it could never destroy the Ring itself. Narsil and the King are one, like Excalibur. And the parallels between Excalibur and Narsil are very powerful.

Glamdring, while a distinguished history, seems to have a more singular and less important purpose: vengeance for the sack of Gondolin, particularly against Balrogs.

Anglachel is for me the most fascinating because it is the only one that's black, and the only one that we know of that speaks. I think that it was doing Morgoth's bidding when it "killed" Beleg, Brandir and Turin -- it was never under Morgoth's sway. Tolkien talks about the sword still having the dark heart of Ëol in it, and he was not a servant of Morgoth. He was his own master in a way, one who had evil designs, perhaps, but was not bound to a Dark Lord. (He would be classified with having a "chaotic-evil" alignment.) Anglachel clearly had no love for the creatures of Morgoth, nor did it particulary love Turin, or even its first master, Beleg, though it was bent on revenge for his death. It was a nasty, but effective sword that didn't seem to discriminate among whom it slew.

Would it have served Ëol differently (ie more obediently)? That's an interesting question. Remember that he was essentially forced to give up Anglachel as payment for dwelling outside Doriath -- a tax or tribute, if you will -- to Thingol, whom Ëol openly disliked. But he chose to keep its twin, Anguirel. The Dark Elf may have realized that Anglachel would have been less pliant and controllable, so he less grudginly parted with it. Or, he placed a curse on the sword before surrenduring it.

There may be some clue in the names of these two swords, but I can't tell what it is yet...
Anglachel = "Iron of the Flaming Star"
Anguirel = "Iron of the Burning Star" (I think that's what it means)

Anguirel was stolen by Ëol's son Maeglin as he and his mother fled from him, before his father could have cursed it. But Maegin was ultimately a Servant of Morgoth, and that may have corrupted his sword as well, but after the sack of Gondolin, it disappears from the story. I like to think it somehow made its way into Sauron's hands who reforged is as the Morgul blade wielded by the Witch King of Angmar in the Third Age. But that's my own little pet hypothesis...

Prancing Pony