Being slightly more specific about what works on the series, one can watch the (and I'll be using season numbers following the new show's numbering not the old one) season one episode "Dalek" and be mesmerized at what this evil creature is, and feel for The Doctor who clearly feels more strongly about the evil the creature represents than he has for anything else all season. One can further sympathize with Rose's position that the creature deserves a chance. Even if you've seen the old series, you might think that in this new world the Daleks deserve a chance – the final Dalek and the final Time Lord, maybe they should be given a go. And, beyond that, there's a classic Cyberman head seen at the beginning of the episode, and though the Doctor doesn't explain who the Cybermen are, he does state that it's an old enemy which is enough for new viewers, while old ones will remember The Doctor's past brushes with them.
That first season of the new series adheres to the old series in one way that later seasons don't – its look. It cannot be denied that as the new show progresses the sets and costumes are more detailed, and the graphics far more realistic, but that first season lags behind the others by so much that even the second season (David Tennant's first) appears to be a quantum leap forward. It looks as though the series' budget rose dramatically or as though someone behind the scenes decided that maybe, just maybe, they didn't have to adhere to the—forgive me—at times schlocky aesthetic of the original.
As for the first Tennant season, it is remarkable. Christopher Eccleston is a fun Doctor and someone who I wish had played the character for a longer time, but watching Tennant is just a wonder. He throws himself into the role so fully that it is hard to watch and not instantly think this is who The Doctor truly is and should be forever more. To some extent it is because the show looks far more current (and far less like a cheap late '80s, early '90s series), but more it is that the writing is more crisp, the plots more deep, and Rose Tyler really comes into her own following the Bad Wolf storyline. But, the biggest reason, is that David Tennant is just so incredibly charismatic.
When Matt Smith appears, the series undergoes another dramatic shift. Some of this no doubt is due to Russell T Davies leaving as showrunner and Steven Moffat stepping in, and some is due the new Doctor. Smith, as I've said before, appears less able to handle the gravitas The Doctor often needs (he is a destroyer of civilizations, including his own). That is replaced with comedy and the show loses something because of it. Additionally, the show's tendency to go for "special" episodes rather than more "regular" ones hurts it and our understanding of The Doctor. It can't be said that any of The Doctor's life is regular, but losing the typical days in favor of special ones doesn't do any favors to Smith or the series.