As the category of smooth manifolds is a full subcategory of the category of diffeological spaces the answer is yes they are still exotic.
Edit: let me add that the forgetful functor $U$ from diffeological spaces to topological spaces given by the $D$-topology has an adjoint let call it $Sm$. If you take a topological space $X$ it has a diffeology where the plots are given by all continuous maps from numerical domains to $X$. If you begin with a non-empty smooth manifold $M$ of $dim>0$ viewed as a diffeological space then you can notice that $Sm(U(M))$ are not isomorphic. This is not surprising of course.
Let me also add that even if you can put plenty of diffeological structures on a topological space, actually we have no classification results nor surgery techniques available in the diffeological world. People are developping homotopy theory for diffeological spaces (Enxin Wu's Phd thesis is a very good place to look at) maybe one day these techniques will be helpful in order to understand our "classical smooth manifolds". Thus actually exotic spheres are not easier to build nor to recognize in the diffeological world.