Change of speed but no change of size after OpenGL Perspective Projection
When you draw a point with GL_POINTS, it's not a 3D object with a size in world-space: its size is a fixed number of pixels. Because of that, the size of the point is independent of the distance from the camera.

If you want to draw dots whose size depends on distance, you either need to compute the perspective yourself and change the point size for each point (not recommended, because state changes like that are expensive); or use another method to draw them. For instance, you might draw a quad and use the fragment shader to make it a circle (e.g. by discarding the fragment if the world co-ordinate was more than a particular distance from the centre of the quad, or by using a transparent texture with a circle in it).