The new one is not "the same show." That show folded back in the 20th century. nuWho is an entirely different production, much more expensive and elaborate and successfully directed at a wider audience.
I see what you mean, and there's certainly a case to be made for both points of view. In many ways it isn't the same show, certainly from a production standpoint. If you look at the original 26-year run, though, the Hartnell era and the McCoy era aren't "the same show", despite having a continuous run. There's no uniform style and approach to the original run, so while the series post-2005 is undoubtedly very different in many respects, it's as different from the Tom Baker era as that is from the Troughton era and so forth.
So you have the early Hartnell stories that were initially intended to be partly-educational, the action adventure of the Troughton years, the
Quatermass-inspired Earthbound Pertwee years, the Hammer-esque Gothic horror of early Tom Baker into the characteristic Douglas Adams stories of his later era, the hard SF of Davison etc. until you get to the more postmodern slightly
Buffy-esque Eccleston/Tennant years and now the new Tim Burton-inspired dark fairytale of the Matt Smith stories.
To me, despite the differences in production techniques, pacing, acting style and so forth, to me it is all the same show as
Doctor Who has never been entirely one thing, consistently. The "old show" may have folded back in the 20th century, but it endured for a quarter of it and ended only because it was allowed to stagnate, and it returned in the 21st because certain people at the BBC who had enjoyed it as children when it was at its peak were now of the age where they were in the position to recommission it. They could see that it had a place in the modern televisual landscape, and of course it did; it's the ultimate format for a TV show.
(YMMV).
