A skyhook uses a shorter tether than a space elevator. My estimate for a skyhook is a 4000 km tether, whereas a space elevator needs 100,000 km tether. A shorter tether is more feasible than a long one because of strength issues. Materials are not yet available for a space elevator, but a skyhook just might be possible with adequate quality control technologies employed during manufacture to ensure maximum tensile strength and uniform elasticity are built into every inch of its length.
The tether itself is the main technological issue. A simple string won't do. Even a tether with a tapered thickness isn't good enough. Why? Because of the risk of micrometeorite impacts which could sever the line. During atmospheric entry the portion of tether that encounters atmospheric drag should be a tight cylinder (like a conventional rope). But the majority of the tether that is always above the atmosphere, more than 3800 km length, is exposed to vacuum continuously, and to space debris impacts.
The entire length of the tether must have a cross-sectional area that varies - thick near the root, and thin near the tip. But in space even a 4 cm-thick tether can be completely severed by a 5-cm space rock. The solution is to fashion the tether into a hollow tube at least 2-3 meters in diameter. That way even a 30-cm space rock can pass through it and compromise only a fraction of its cross-sectional area. The tether would remain functional, at least long enough to complete any launch of a payload that is in progress.
The tube tether must be repaired as well. That means removing severed fibers from the tether and replacing them with new ones. The most obvious way of doing this is to retract the tether, and perhaps change it out with a new one during a shuttle re-supply mission. But we could instead fashion the tether so that it has load-bearing nodes spaced every 400 km along its length. That way at most only a 400 km long fiber would have to be replaced. A robotic climber could do that, or a partial tether retract with repairs carried out with robotic arms could be accomplished. What's more, we don't want 400 km-long fibers hanging loose causing havoc, so quite frequently there should be non-load-bearing nodes built into the tube tether. These would essentially hold the severed fibers in place, allowing perhaps only a few kilometers to dangle around.
This is complicated, but there are so few people working on the problem. If you look at the space shuttle, which is considered to be the most complex machine ever built, you will find thousands of people dedicating their careers to making it work. If you throw those kind of resources at something as seemingly complex as a skyhook you will probably see it work successfully.
But back to the space elevator versus skyhook debate....
What if you wanted to build a space elevator on the moon? Or Venus? These worlds have rotation periods ranging from a month to a year. There isn't enough rotation to keep a space elevator up, since a space elevator requires a geosynchronous position, AND rotation to keep the tether taut. Also, a space elevator does not deliver a payload to orbital velocity *unless* it is released above geosynchronous altitude. And with a space elevator it would take a few weeks at least to get there. A skyhook delivers a payload to orbit in a few hours. It's sole purpose is to get the payloads above the atmosphere. Once there it can employ super-efficient electrodynamic tethers or ion engines which cannot be used to generate high thrust for launch. Those super-efficient engines can take the payload anywhere, to geosynchronous or beyond, nearly as efficiently as a space elevator.
But the main problem with a space elevator is tether strength. There are a few possible candidates for a skyhook, but there are none currently available for a space elevator. The only one being proposed is carbon nanotubes, and a recent announcement by Chinese researchers who managed to draw 30 cm-long nanotube strands, were only able to do so by simple entanglement of filaments. The resulting strand was over a million times weaker than the constituent fibers. There was no chemical bonding taking place between the nanotubes that would create superstrong tether.
As for THE CUBE, it is so-named because it is shaped like a cube, and remains cube-shaped even when it changes size.