From the perspective of set theory, the line and the plane are the same: they are both infinite sets, and the same size. But we have many other ways to understand the line and the plane (and 3-dimensional space, and 4-dimensional space…).
A topologist will tell you that a line and plane are different for the following reason: if you take one point out of a line, you get two disconnected pieces; if you take one point out of a plane, the result is still one connected piece.
A geometer will tell you that the line is one-dimensional and the plane is two-dimensional; it takes one number to describe a point in the former, and two for a point in the latter. But, at the basic level of sets, these two mathematical objects are the same size.