Is spacetime curvature and the shape of the planet is the same?

The example of people walking on the surface of the Earth is often used to demonstrate how curvature can cause an apparent force between two objects. I have used this analogy myself for example in my answer to When objects fall along geodesic paths of curved space-time, why is there no force acting on them?

But it is important to understand that this is just an analogy. The curvature of spacetime is not the same as the curvature of a sphere. There are some important differences:

1. the curvature of spacetime is intrinsic not extrinsic

2. spacetime is a pseudo-riemannian manifold while a sphere is a Riemannian manifold

3. for spacetime we have to consider curvature in the time dimension as well as in the space dimensions

I'm afraid explaining what these statements mean would get very complicated very quickly, but of all of these (3) is probably the most important difference. On the sphere if the two observers stood still they would not approach each other. But we are all moving through time (at about one second per second) and under most circumstances it's this motion through time that causes what we see as acceleration due to gravity. This motion through time is completely missing from the analogy of the sphere, and it's why the analogy breaks down when we look at it too closely.