In US English, “The story is about the childhood of Tom” is more natural. Also, “Tom’s childhood” is more natural sounding than “the childhood of Tom.”

For a general sentence, "The story is about Tom's childhood." sounds a little better. In the case that there was more information following the first statement, for example, "The story tells us about Tom's childhood and the hardships he faced at the time." When you add "about" you can use your original order, it just feels like there's likely to be more information to follow.

Maybe just like the story tells us the childhood of Tom Or should we say The story is about the childhood of Tom? Which one is more authentic?

Can you provide more context?