Q:

Why is *vertical* autorotation in a helicopter not recommended?

Helicopters can autorotate back to earth in case of emergency. The FAA recommends a flight profile similar to the following procedure:

Notice that in section (2) there is forward speed: vertical autorotation is not recommended. Why is that?

A:

When the engine fails in a single engine helicopter it can autorotate back to earth: it gives up potential energy from reducing height to keep the rotor turning, which then continues to provide lift. Autorotation can  take place straight downwards in vertical flight, but the general recommendation is to maintain forward speed, like a fixed wing gliding: the rate of descent is much lower if there is forward speed.
 Image source
We associate helicopters with hovering, but this is a strenuous effort. Even a bit of forward velocity generates translational lift, which makes it much easier for the rotor to generate lift: 

The forward inflow tilts the lift vector forward, decreasing induced drag and blade torque, and increasing lift. Or stated in a different way: the airstream is flowing in already, and does not need to be accelerated from zero. The rotor now starts to work more like a fixed wing does.
The airstream through the rotor experiences less interference from the fuselage.

The effect of forward velocity on auto rotational rate of descent is depicted in a graph in this book:

Going back to vertical autorotation: the vertical upward airstream needs to go through the rotor fast enough to reach the windmilling state, so that lift is created by decelerating the air passing through it. If we compute the lift component per area of a vertically autorotating rotor, it is comparable to a $C_D$ value of 1.1 to 1.2 referenced to the rotor area. According to this source:

a flat plate has a $C_D$ of 1.28
a parachute has a $C_D$ of 1.4. 

So in vertical descent, the auto-rotating rotor is almost as good as a parachute of the same area - it’s just that the parachute size is a bit small for the typical payload of a helicopter, resulting in vertical descent speeds of between 3,600 to 6,200 ft/min. By way of comparison: 

Air France 447 fell to the ground with a vertical descent rate of 10,000 ft/min. 
Captain Sullenberger's plane took four minutes to reach the Hudson in a proper glide, from 2,060 ft altitude = average of 500 ft/min.

Two factors make autorotation much more survivable though:

The descent rate can be “flared” by pulling the cyclic back (exchanging forward velocity for more rotor kinetic energy) and then collective up just before ground contact, slowing down the rotor and increasing lift. The pilot only has one shot at this. 
The descent rate is a lot less if autorotation is done at a forward speed,   the best one being the optimum climb speed.

Note that in a vertical descend there is no pullback of the cyclic, only increase of collective - plus the rate of descent is higher. A higher speed and fewer means to brake, that is not recommended!
A side note
The windmilling rotor does not turn the other way! The incoming air from underneath tilts the lift vector forward, like in a glider. Part of the rotor disk is driven by the aeroforces into the same direction as it was when still under power.
 Image source

A:

There are three good reasons to avoid Vertical Autorotation.
First, Vertical Autorotation is way more likely to lead to a Vortex Ring condition where the air exiting the rotor disk gets recirculated back through the rotor disk.  The recirculated air in the Vortex Ring then limits the amount of airflow/energy available for conversion to Rotor Disk rotational velocity which is critical to successfully completing that part of the Auto Rotation maneuver essential to survival (the flare up to touchdown). 
Second, the forward velocity increases the amount of energy available for conversion to rotor disk speed. 
Third, forward velocity allows landing shock to be shared between different harnesses and seat shock absorbers thus decreasing the load on any particular axis.

A:

Forgetting THEORY... a vertical autorotation IS possible and can be accomplished safely. To answer this question put to me by my Co-pilot that day in Vietnam, I took our H-Model Huey up to 5,000' and confirmed hover. I lowered the collective and chopped the throttle to split the needles and monitored rotor RPM to ensure no over-speed while keeping the cyclic centered for true vertical. I used collective to stabilize our sink rate. Once satisfied that the maneuver was safe, I recovered at about 1,000' by pushing the cyclic forward and recovering power. 
We experienced no issues with vortex ring theory or any indication that this maneuver was dangerous. Had I needed to actually put it down, I would have initiated a forward cyclic aspect in order to perform a standard flare and set-down once below that 1,000' I chose. In Army flight school in the 1960's, we practiced every possible aspect of auto-rotations ALWAYS to the ground with NO power recovery at 50' nonsense. I personally believe it's irresponsible and insane to not train a helicopter pilot for full ground contact autos. What? Is God going to suddenly show up and help in that last 50' or so? 
SIMS are NOT an acceptable substitute for actual practice. We did hovering autos, run-on autos, 180° and 360° autos, full-speed deceleration autos, but my all-time favorite was called "Spot Autos." The IP picked a spot on the runway - usually one of the numbers and I initiated an auto from pattern altitude and speed on the runway heading noting my steep glide angle through the chin bubble. 
Keeping collective down and using only cyclic, I flared the chopper to an almost dead stop - closely monitoring rotor rpm and pulling collective when needed to prevent over-speed. Cyclic was pushed forward to start the wonderfully fast descent again - oh wee - we could get at least two and sometimes three flares on the way down - the last one was to a stop right on the number chosen and pull what was in essence a hovering auto to land. 
We always completed all our autos to the ground with all of these maneuvers and they were truly FUN to do! Our IP's defined expertise and competence and did an excellent job of instilling those same characteristics into all of us, knowing that where we were headed we would need every training advantage we could handle. My two cents (1,955 combat helicopter flight hours in Vietnam slicks and guns).        

