Since you create a motion path in the scene and not in the Channel Editor, you do not have to be positioned at a specific keyframe to move the vertices of a motion path. You can move vertices and tangent handles for any keyframe in the animation.
Moving a locked vertex on the spline changes the position of the object at that frame. A locked keyframe on the speed curve can only be moved along the horizontal axis (frame number). This changes the frame where the animation reaches a particular point on the spline, but not the shape of the spline. Thus only the animation timing is affected.
To move a vertex on a motion path:
To move a keyframe on the speed curve:
To move the speed curve in the Animation Curve window: