1. the quality of being graceful and stylish in appearance or manner; style.
“a slender woman with grace and elegance”
2. the quality of being pleasingly ingenious and simple; neatness.
“the simplicity and elegance of the solution”
So not only there is an overlapping of meaning between the nouns elegance and style, but data say that elegance is progressively going out of fashion, subbed by the more generic and appealing almost-synonim.
Still, especially in (magazine) design, style and elegance are two extremely different ideas, almost opposite.
Achieving elegant results requires expertise, refinement and, usually, a lot of time. To be stylish you need a good knowledge of what is hot at the moment and great tech skills to reproduce it. Neither of the two is an easy job, but apparently elegant designs – by definition pleasingly ingenious and simple – tend to be long-lasting and some of them become timeless. Which, I think, is what we all should aim at.