Why do semiconductors remain neutral outside the depletion region? Why there is a sharp cut off of the charged region outside the depletion region, like on this image?
For example why don't electrons on the conduction band in the n-type side rush towards the positively charged area making the whole piece positively charged somewhat, not just at the area near the depletion region?
The source of the confusion is that I know if you charge up a regular conductors the internal currents will uniformly distribute the charge along the whole piece, while insulators are only locally charged up, since they cannot carry current. 
Semiconductors here seem to act like insulators, but diodes do carry current when used. How?
 A: I think the graphic you're looking at is a simplification.  The positive and negative ends of the charge regions are probably a bit more trapezoidal (sloped) 
than the sharp discontinuities represented there for simplicity.  After all, the color representation above is a bit fuzzy at the edges.
A: The negative charges in the n type side are repelled by the negative ions in the p type region and positive charges in the p type side are repelled by the positive ions in the n type region in the equilibrium condition of diode. That's why charges are unable to travel across the pn junction. I hope you know about the n type and p type materials.
A: Look up the depletion region approximation. Which by the way is actually pretty good, especially when the junction is revered biased.
When it is forward biased, or one side is highly doped sometimes it is not as good and it would be better to work out the solution numerically. Then you could find that the edges are not sharp.
Part of the reason the edges are sharp when you make the approximation is that the carrier concentrations are pretty large and the doping is supposed to be uniform. Then assumption is that there are no  free carriers since they get swept out by the electric field.
I think the other thing to consider in answering your question is that you end up balancing drift and diffusion currents to maintain charge neutrality.
A: From your response to @JonCuster:

The question is. when the pieces are in equilibrium why don't the remaining charge carriers distribute themselves uniformly in each half.

The remaining charge carriers within the depletion region are not electrons but semiconductor ions (both positive and negative) locked into their positions within the crystal so they can't move. If the lattice ions could move, then they would redistribute themselves to eliminate or minimize the electric field across the junction.
