Different definitions of the Fermat curve

Question

I am using the definition $X^n+Y^n-Z^n=0$, although I know that the apparently similar alternative $X^n+Y^n+Z^n=0$ is in common use. I wonder if the two are isomorphic? I think they are, otherwise people wouldn't use both definitions with such nonchalance. But then a $\pm$ sign can make a huge difference over fields that are not algebraically closed: think of the empty, projective $\mathbb{R}-$variety $X^2+Y^2=0$ versus the non-empty, non-irreducible $X^2-Y^2=0$.