There isn't really an explanation for this.
Power Gem is notable for being exclusive to Pokémon with designs that incorporate gemlike features, thus being distributed among Pokémon independent of type. Despite this Diancie is unable to learn Power Gem for unknown reasons.
What I find most shocking is that Carbink can learn it, yet Diancie can't for some reason, especially considering that:
>Diancie is born from a Carbink that undergoes a mutation. It can instantly create diamonds by compressing the carbon in the air; these diamonds can then be used to defend itself.
The reason why Diancie doesn't get Power Gem could be because Diancie got Diamond Storm as it's signature move, which is a more powerful, STAB Rock type move.
What's strange is that despite not being able to learn Power Gem, Diancie has been seen using Power Gem on the Pokémon X and Y official website inside a cave.
>Despite not being able to learn it, Diancie is seen using Power Gem on X and Y's official website.
However this turned out to be Diancie actually using Nature Power, as Nature Power becomes Power Gem when used inside a cave in Generation VI. (Credit to user Shaishai8 for pointing that out to me.)
Here is the image of Diancie using Power Gem through Nature Power on the Pokémon X and Y official website:
TL;DR:
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