My basic theory of the differences of teleportational abilities is as follows.
In the original series, we know that the Tomorrow People require jaunting belts or bands, in order to teleport any real distance. I also assume that whatever device John developed to eliminate hyperspatial zonal shift (the strange lights apparent upon jaunting exhibited prior to The Blue and the Green) is somehow built into the belt, and that there are crystals of the type which John and Kenny use in Slaves of Jedikiah to communicate with TIM contained in the belt. The question is why do they need the belts/bands (I will just refer to them as belts) in order to travel any real distance.
There are two basic theories, as I see it. Either or both could be true. These theories are:
We know that the spaceship has been there for thousands of years (Origin Story). We know it has been waiting for the Tomorrow People. We also know that it has been buried for a long time, possibly due to earthquakes.
It appears to me that the large light on the spaceship could very easily be like TIM - a biotronic computer, although perhaps a more primitive form, as it cannot communicate by speech (or perhaps it is damaged).
Let us suppose that at some time in the distant past, some alien race left the ship on the island as a way of helping future Tomorrow People. We know that the ship is able to communicate with the Tomorrow People, and tell them what they are. We also know (from The Visitor) that John, Carol, and Kenny had to travel off planet to learn what they were.
This alien race created the beacon in order to give the new Tomorrow People a safe place to teleport to, and in order that they could find out what they are. However let us assume that at some time between activation and 1970 (the year of John's break out), the ship was damaged, perhaps by an earthquake, that also half submerged it.
The apparatus, the beacon, failed. John broke out, so did all the other Tomorrow People, up until Andrew without its benefits, and without any knowledge of its existence. At some point they left the planet (read Exodus for my theory).
But the ship was intelligent and had the ability to repair itself. It began to do so. By 1992 when Adam broke out, the beacon was once again working, at least to an extent. It drew him near it, but not too near.
Now lets assume that there was one other factor in the beacons construction. It created some sort of energy lattice across the planet. The idea of transferring energy through the air is not fiction, itís a valid scientific principle (dating back to Tesla). Now this energy could be available to the Tomorrow People either as a boost to their teleportational abilities, either for general long distance teleports, or when power is seriously needed - Ami in Monsoon Man, to bring Adam and Megabyte to her, or in The Rameses Connection when the other rescue Adam and Millicent. Alternatively it could be used as a carrier wave which transmits a variable signal which allows the TP to navigate.
Therefore the new Tomorrow People do not need to wear belts/bands to teleport. Although maybe if they had them they wouldnít disappear and reappear in a flash of light, and maybe they could talk to TIM.
Also the spaceship could have been continuing to repair itself all through the new series, just getting the location of the beacon right in order for Jade to teleport without falling into the sea, during The Living Stones.