Variety vs. Manifold

In the ambit of differential geometry the aim is to study smooth manifolds. Why the objects studied in algebraic geometry are called algebraic varieties and not for example algebraic manifolds? I am a fan of the uniqueness of mathematics, so I think that these differences of terminology or notation may mislead the student.

Solution 1:

A variety does not qualify as a manifold for more reasons other than smoothness. For example the $xy$-plane union the $z$-axis is a variety. But, there isn't even a well-defined dimension there. You would need a sufficiently broad definition of manifold to include varieties that are not smooth and don't have a dimension. At that point, the word "manifold" would not be very useful.

Solution 2:

Many algebraic varieties are not manifolds. For example, the coordinate axes in $\mathbb{R}^2$ are an algebraic variety, but not a manifold because it isn't locally homeomorphic to $\mathbb{R}$ at the origin.

Edit: Thank you to Robert for pointing out that the issue here isn't smoothness, I think I was slightly on autopilot. You can also get algebraic varieties which are non-smooth even though they are manifolds, such as this one. But as has been pointed out already, depending on your definition of manifold this may be fine anyway.