UV space
A UVMap basically assigns a (u,v) coordinate to every point on a mesh. You can think of this as a function that assigns to every point on a mesh, a point in a plane. That plane is UV space. It is not a part of the XYZ space the mesh exists in. There's only one UV space for the whole mesh.
Tangent space
Every point on a mesh has a different tangent space. The tangent space can be thought of as all the arrows rooted at that point that lie in the tangent plane (ie. that are perpendicular to the normal). Sometimes a third dimension (the direction of the normal) is added. Unlike UV space you can think of the tangent space as embedded in the same XYZ space the mesh exists in. The zero point in the tangent space at point p would correspond to p itself.