Yeah, in the cutscenes, the horn grows and shrinks for the purpose of dramatic effect. But if you play the game this way, the horn grows and stays that way.
Other than that, I don't know why you'd do it. Of course, some fans believe it will unlock some hidden mission or something, but you know...
Edit: I'm pulling this from another forum. In case anyone else is interested in growing their horn.
1- Normal Snake: starting level; basic appearance of Venom Snake.
2- Horn Snake: it has a longer horn than Normal Snake (
http://imgur.com/EbBrNdP). In order to increase your Demon Level to 2 for the first time, you must accumulate 20,000 Demon Points.
3- Demon Snake: it has the longest horn and Snake is always covered in blood (
http://imgur.com/RwjC9lQ). In order to increase your Demon Level to 3 for the first time, you must accumulate 50,000 Demon Points.
I had no idea, man.