Why is Byrne vs. Fischer (1956) regarded as the "Game of the Century"? (This is the game in question.) Is it because it was played by a 13-year-old Bobby Fischer?  Would it have been different if it were played by a much older player?  Who labeled it the "game of the century" and did this game between Tal and Smyslov (1959) miss being the "game of the century" by 3 years?
The "Game of the Century" was a term coined by Hans Kmoch in Chess Review. In short it was a bit of sensationalism. While Fischer did show a talent for combinations and an impressive sacrifice, the real importance of the game was the story of a leading chess master vs a phenomenal prodigy about to begin his historic rise.

Still, it's helpful to note what David Levy said: "In my opinion, however, the game, attractive though it is, can in no way be held in such high exaltation. It was merely the fact that the combination was created by a thirteen-old boy that brought the game so much publicity. Had it been played in the Barnet League between two sixty-year-old men it is doubtful whether it would have been considered worthy of publication".

As for the game between Tal and Smyslov, the similarities in age are easy to spot, but the chess playing doesn't really compare. 

The full game and commentary can be found over at Wikipedia.