Mystics are (most-likely) humans, so their magical abilities would come from the same place as a human's magical ability (which is implied to be an effect of Lavoid-Radiation, aka the Frozen Flame). Spekkio groups mystics in with humans when he is talking about magic:
You are strong of will...! / Hm?
That's why the Old One let you / You guys have it.
through. / Power of the heart......
/ I get it, that's why the old man outside let you
/ in here.
Long before you were born... / Long before you guys were born......
...there was a kingdom where / There was a kingdom that prospered by magic.
magic flourished. /
Everyone there could use it! / Everyone in that world used magic.
But in time, people began to abuse / But that country grew addicted to magical power
their powers. It got so bad that no / and was destroyed......
one was allowed to use magic / After that, people became unable to use magic.
except wizards. / Except for the Demons anyway.
But you have it...determination, I / But you guys've got it. Strength of the heart.
mean! /
Magic needs power of the heart. /
It needs inner strength. / Magic's the heart's strength, power.
Mystics might be different than humans, but if so, then Spekkio's comment regarding them is very non-sequitur.
Humans, as well as mystics, seem capable of drawing energy directly from the planet. The sun stone can't be powered by Lavos as there is a distinction made between the elemental energy that Zealeans used to use and the Lavos-based power they switched to. As Dalton clearly was able to use magic after the fall of zeal, the Zealeans who claimed they could no longer use magic seem to be referencing personal limitations, not that the ability was actually lost. Possibly they just didn't remember how to tap into the planet anymore.
As for why Mystics look different, meh, I'm not convinced that they
are different. Humans are a bit of an anomaly in that we are a very homogenous species (there was a bottleneck in our ancestry. All modern humans came from a single small group of early humans; if I recall correctly, that group was around 1,000 individuals, though I may be forgetting a zero).
Consider the wide variety of dogs:
http://en.wikipedia.org/wiki/List_of_dog_breedsDifferent "breeds" of a species can take on radically different morphologies and still be inter-related. As such, it seems to me that since the game treats them as a single species, one would need some specific evidence to argue that they are in fact separate races.