Variety vs Manifold (2 Major Differences)

There are 2 major differences between a variety and a manifold.

1. A manifold looks like R^n if you keep zooming in. A variety can have corners, cusps, crossings (X marks), etc. You cannot zoom in and get rid of them. If a variety does not have them we call it a smooth variety. All smooth varieties are manifolds.
2. An entire variety is described by a single polynomial. This can sometimes apply to a manifold, but often not the case. Maybe only some patches can be described by polynomials. But, in any case, often an entire manifold cannot be described by a single polynomial.