In the ideal case where there is no friction and no perturbations and the top starts to spin in a perfectly vertical alignment, the two configurations (inverted or not) of the top are completely identical.
However, once you have the top start rotating with a tilt from the vertical axis, or consider perturbations that will tilt it even if it wasn't, then the top will undergo precession. The main difference now becomes that when the top is on a point, precession will not change the point of contact with the ground, however when its on the other side, precession will automatically make the point of contact with the ground change! Now if you add friction, and remember that these changing new points of contact are rotating with the rotation speed not the precession speed, you can understand how the spinning is destroyed.
Notice how the spinning top on its flat area spins the best when the precession speed and rotation speed are roughly the same, consistent with our discussion.