Both TNG and New Who are sequels and
continuations of their predecessor series. They are both set in the same universe and continuity of the previous shows. TNG and other Trek's, had numerous references to each-other and the TOS crew, as well as cameos and guest appearances by major and minor characters, so there is no doubt that they are all set in the universe. The DS9 episode 'Trials and Tibble-ations' is a good example of how they are all the same world. The new Doctor Who is different, in that it is both set in the same continuity as the 63 - 89 show, but features the same lead character.
Major and minor characters like Kirk and Spock, have made special appearances alongside the other crews, but each series generally has a different main cast. This doesn't necessarily make either any more or less of a 'sequel' as they are both new and separate productions from the original, and continue the story and world of their originals, but because Who has the same main lead, it is perhaps a bit closer to its original in terms of the story. From a production standpoint, TNG is a bit closer to the original, because it was filmed in the same place and had the same creative lead. Even the new Star Trek movies are part of the larger continuity, because they don't ignore the prime universe, so they aren't a total reboot.