Yes, although there are still many similarities and the general concept of the series remains the same. The original series began with the Tomorrow People comprising John, Carol, Kenny, and their new recruit Stephen. John is the eldest member of the team and he was the one responsible for building TIM, a biotronic, sentient computer which aids the team and can augment their powers. In the original version, the team still represent the next phase of human evolution and are called homo-superior, in contrast to the rest of the human race who are homo-sapiens. Their powers included teleportation (called "jaunting"), telekinesis, and telepathy. "Jaunting" over long distances was done with the aid of a belt (later a bracelet). The team's goal was to assist other Tomorrow People as they "break-out" (i.e. when their special powers become active, usually in adolescence), and they operated from an underground base called The Lab which was situated in an abandoned part of London's Underground Rail Network. As well as Earth-bound adventures, the series featured several stories set in space and on alien worlds (the team were affiliated with a Galactic Federation which was analogous to the United Nations). New Tomorrow People joined the team on a regular basis, though there was never more than four or five team members at any one time.
In the US remake, the main characters still include John and Stephen, though Carol is now called Cara and Russell appears to be the closest analog to Kenny. Again, Stephen is a new recruit who joins the team to assist others as they "break out", though we learn Stephen's father was also a Tomorrow Person which is not something the original series did. The team still operate from an underground subway base, though it is not referred to as "The Lab" as in the original, and there are far more Tomorrow People who inhabit the base. TIM is also present in the new series, and was again built by John, though looks different to the original version. Again, the Tomorrow People have psionic powers, though teleportation is not referred to as jaunting in this version and does not require a belt. Newer powers are also seen, such as Stephen's ability to stop time. So far in this version, all of the Tomorrow People's activities have been set on Earth and there is no indication of a Galactic Federation or even extra-terrestrial life.
In both versions, the Tomorrow People have an adversary called Jedikiah. In the original version, Jedikiah was a shape-shifting alien robot who later took on human form. In the US remake, he is the uncle of Stephen and is a high-ranking official in an organisation called Ultra which aims to hunt down all of the Tomorrow People as they are perceived as a threat to homo-sapiens.