Enchantment or advancement.

[Hookbill the Goomba]

Perhaps this all comes down to power, or perhaps, perceived power.

Having a gun can, as Mnemosyne pointed out, give almost anyone power. A sword, while still dangerous in any hands, is always more dangerous in the hands of a trained swordsman. (To an extent, someone who has practice with a gun will be better at aiming, to be sure, but a gun in the hands of a novice can still create fear, and that's the main point of it).

But where enchantment is concerned, I think it goes beyond swords. It was the thought of Beren that got me on this line of thinking. What stops him from being killed on his journey towards Doriath? What makes Thingol think twice about killing him? His sword? No. The ring of Felagund.
We are never told if this ring has magical powers, exactly, but it certainly has some power. Perhaps it is of a different kind. The respect or memory they held for its previous wearer make it more than a ring. Although it cannot kill, it stops them in their tracks. Few blades could do that.

If I may take an example from Doctor Who.
In the Ninth Doctor's final episode he confronts The Daleks. The Doctor has no weapons or means of stopping them. But the Daleks are still terrified of him. Not because of what he has in terms of threat, as such, but because of their memory of him and what he has done to them in the past. The oncoming storm.

Now, back to Middle Earth.

A similar thing may be seen with Anduril. More than just an elvish blade, it is THE sword, the one that cut the Ring off Sauron's hand. Moreover, it was wielded by Aragorn's forefathers. It aided in Sauron's first fall, it should aid in his final fall. From a narrative stance, this is quite appropriate.

Another good example could be the Malorn tree that Sam plants in Hobbiton. It is special not only in its uniqueness (the only Malorn west of the mountains, east of the sea), but in the memory it instils in Sam. Of Lorien and Galadriel. I'm sure Gimli would have had a fair few things to say about it.