Manifold locally looks like a open set but not as a euclidean space?

Question

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define tangent vectors in general sense. My doubt is even in $\mathbb{R^n}$, we check differentiablity only in an open set near a point rather than the whole space. So, why are we concerned about the manifold not looking like Euclidean space? Also explain difference between M is locally an open set and M is locally an Euclidean space. This paragraph is from the book i am reading.