(I haven't played 13th Age, but this sounds pretty system agnostic.)
Start the campaign with no usage of the unnatural magic, and have it be introduced with, for example, the first "mini-boss." Increase use of it after that point. To further the difference, give the "unnatural" magic a different name, like witchcraft, demon-magic, old-magic, depending on the feel you want.
In order for it to feel "unnatural", the "natural" magic has to be established first. Introduce them well to the common-place setting of the ritualistic magic of weather, fire for forges, etc. Note to the players that the magic is part of everyday life.
Introduce the unnatural magic in an environment that has all of the features you want the magic to have features of: dark, creepy, scary. A deep dungeon of demons or abandoned temple of old-magic worshipers are some ideas.
When the magic itself is described, ensure you mention what is happening when the magic is cast. For example:
"The cultist summons a ball of unnatural fire from the beyond! As it manifests, you hear the screams of a hundred souls before it hits the wall near you!"
Other examples could involve bad things happening to the caster of the magic, such as draining life force or grotesque side effects. Keep it creepy!