Differential Geometry: Understanding the Affine Tangent Plane at a Point

[knowswhale]

I'm a high school physics teacher trying to get a handle on differential geometry so please make explanations as simple as possible.

What is the difference between the tangent space at p and the affine tangent plane at p?

 

[WWGD]

Maybe if you give us a source ; the place where you saw it, or some other context, we may be better able to help you.