Chinese and Japanese dragons are always portrayed with a long tendril appended to each side of the snout or muzzle. Looks kinda like a mustache, but not much. Where did they get that idea from, the tentacled snake? Probably not, it’s not native to either country, it’s found in Thailand, Vietnam, etc. Are there any theories?
To me they always looked like tendrils from a catfish. There are species of those native to China.
I always thought they were whiskers of some sort, but I got no evidence or facts whatsoever.
From Wikipedia:
I think listedmia is on the right track. While Western dragons are winged lizards, Eastern dragons are more fish-like. Think Miyazaki’s Spirited Away.
Because dragons have those in real life.
Someone had to say it.
The wings are hardly ubiquitous in Western dragon design; while they often appeared on Welsh dragons, the Norse dragon was wingless, as we other European dragons.
I always thought of them as catfish whiskers. One of the first stories I remember about “Asian” dragons was one dragon was the spirit of a river, and he shapeshifted between being a non-corporeal part of the river itself, a big fish in the river, his dragon form, and the form of a young man. I think it was some variant on the “catch and release the magic fish for a prize or reward” story, with a little bit of a love story mixed in there somehow.
Since the dragon was part of the river, and the illustration of him as a fish showed him as a catfish-type of fish (with those same whiskers) then I just figured that’s what it was.
He looked sort of like that as a fish.
The dragon in Spirited Away had fur. Not very fish-like. Not even seal-like, more like a cat’s or dog’s fur.
According to Wiki, the Chinese dragon is a sort of composite creature:
But, nothing about the nose-tendrils.
On this one, the tendrils are erect and project forward.

…more like a cat’s or dog’s fur.
The Chinese Dragon is closely associated with carp. Some, such as the Yu Lung, are literally carp that turn into dragon, others are the descendants of carp and others just have carp features such as scales.
One of the most notable features of koi (oriental carp) are two fleshy barbels forming a moustache-like structureon the upper lip. When the mouth is held open thebarbels point forwards as in BrainGlutton’s picture.
These barbels have commonly been transferred to images of dragons.
But that’s like asking why Bugs Bunny can talk.

But that’s like asking why Bugs Bunny can talk.
Not really, Bugs Bunny can talk because a small handful of his creators said that him talking would work well for their property. Eastern Dragons (and anything else in folklore) have traits collectively decided upon and codified by millions of people over the course of history.
Asking “why” on either of them is a perfectly fine question (there might be a good, interesting production reason why Bugs isn’t a silent character), but asking the question about folklore says important things about the consciousness of the culture it comes from.
Dragons live in the clouds and are associated with rain. Sometimes they’re golden. It’s seemed probable to me for years that dragons are inspired at least in part by lightning, and may be viewed as lightning made flesh. Those tendrils and excresences on its body that resemble those of carp and other creatures are, to my mind, a way of interpreting the branching nature of lightning.
There are other figures in Western art and myth that are similar to this. But the Western dragon seems to be a very different beast.

Bakshi’s Smaug was a dragon with fur. His head looked like a wolf’s.