We call an atlas smooth if each of its transition maps (or change of charts) is smooth (or $C^\infty$). A differential structure is a maximal smooth atlas. A smooth manifold is a topological manifold together with a smooth structure.
These are the definitions of my lecture notes. I am also wondering now why we have not defined what a smooth structure is and if one has to replace smooth structure with differentiable structure instead. In Loring Tu's An Introduction to Manifolds it is also mentioned that it has to be a differentiable structure.
But I tried to compare both definitions on Wikipedia and I can not really tell the crucial difference between these both terms. Both are somehow dealing with maximal atlases.
Could you give me an explanation about the right definition about smooth manifolds? Thank you.