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.

[Mnemosyne]

And let's not forget about the goblins in The Hobbit remembering Glamdring and Orcrist from Gondolin! Talk about long memories!

[Kent2010]

Hookbill

In some cases the names become stories and it is the stories and significances that give them power.

I agree. The name is significant. A name attaches an identity, a story. The Ringwraiths are not named, their personal identity has been lost, and they are in absolute servitude to Sauron. The same can be said about the Mouth of Sauron, his true name has been forgotten and "The Mouth of Sauron" is just a title, a position, not an actual name.

The Ents names (their Entish names) Treebeard says he can't tell Merry and Pippin because his name is constantly growing. The Ents' names are essentially their life stories.

Speaking of 'enchanted' swords I wonder exactly when do these magical weapons get named? Do they get named upon being crafted (was it the Japanese who named their swords?) or were they named after accomplishing a great deed? Was "Narsil" really some special/enchanted blade or was it because of the name, the story, attached to the blade? Names can create stories and those stories can form part of the legend, or the magic.

What about why the Elven Rings were given names but not the other rings of power? 'The One Ring' is afterall a title, not a name.

Lalwende

Or should we bring forwards this general idea of a "fall from grace" here? So in the earlier times everything was better and now all is crap? People used to live in paradise but now they are estranged from that holy or primordial union with God / nature / natural relation with the world... what have you?

I think people do make that assumption and it can be a dangerous one to make. The ideas of "progress," "advancement," "reform,"...etc is what is new, not the belief things were better in the past and now we're all 'falling from grace.' Mass crime, prostitution, scandals, corporate greed, adultery and the whole lot has existed for a long, long, time. It wasn't until about 160-170 years ago when people thought the problems got so bad, reform needed to happen. That's where progress, and the ideas of "reforming the person" to rid the world of its evils, took off.

What's interesting is technology and mass production was seen as the way to get out of the "savage" curses of the past. The first dagguerotypes (I believe about 1820s?) were seen as magicians. Nathaniel Hawthorne's The House of the Seven Gables is I believe an excellent book which tries to argue that through technology, and interesting enough...nature, we can achieve progress and escape the dark, often dirty, past. This argument was the new idea in the world, not society had fallen from its glory days, and needed to be restored to its glory days - Society needed to make its own new glory days.

We get a revolution of ideas through history, and I think WWI brought out a new side to technology and mass production that people never thought was possible. That side left a huge black mark on technology, and using technology for "advancement."

In my opinion it's not technology, the sciences...etc, that is evil, it's how we decide to use it. Didn't Tolkien say something similar about magic in his books? It can be used for healing, preservation, protection, but also domination and destruction.