I thought it was a custom in English to write "it is raining" instead of "rain is falling". Still I don't know why the second construction is wrong. Yet I found a song called Rain is Falling, so I'm confused. Is this construction entirely wrong?
It is raining is how we normally describe the weather on a rainy day. However, that doesn't make rain is falling grammatically incorrect. That construction may not be a common way to describe the weather, but it's not "wrong."
The word rain can be a verb, or a noun; as a noun, it refers collectively to raindrops. Moreover, falling can refer to anything dropping from the sky. Therefore, paratroopers can be falling, drones can be falling, and rain can be falling.
One might regard "rain is falling" to be a more poetic way to talk about the rain; for example, "Rain is falling in Santa Fe" might be considered more poetic than "It is raining in Santa Fe."
Here's another place where I could see a writer using this way of phrasing it:
The firefighters had been battling the wildfires for three days. On Friday morning, their prayers were finally answered; rain was falling as the sun came over the mountains.
Bottom line: Don't confuse "less common" with "incorrect".
There is absolutely nothing wrong with the construct "rain is falling." The listener would know what you were saying (correctly) either way. But it sounds a bit more natural when you are talking about the state of the weather to say, "It is raining."
The word rain describes both an object (the physical rain from the sky) and a state of the weather (it is raining today). So if you are stating what the weather is, it sounds a bit redundant to say it is falling. "It is raining" versus "There is rain outside and it is falling."
The construct seems pretty consistent.
"It is cloudy." — describing the state of the weather.
"There are clouds in the sky." — describing the physical object.
Well, I might find it redundant because RAIN is meant to fall and you cannot say that "The rain is falling." but instead, we should say "It is raining." Another thing, it is grammatically correct that is why some writers use it to their pieces. It is accepted in literature, especially in poems, because we have poetic license.