Though Barnabas Collins is very popular right now thanks to Johnny Depp and I AM a fan of the classic Dark Shadows I will have to say Dracula. Dracula is who inspired the creation of Barnabas Collins by Dan Curtis and Dracula continues to have a great deal more in the way of powers than Barnabas.
I'll make this comparison using literary Dracula and the historical figure he is based on for pre-vampire backstory. For Barnabas I will access all the incarnations of the character for this comparison.
1. Dracula can walk about by day though it exhausts him and he cannot shapeshift by day, whereas Barnabas burns if in direct sunlight in the movie and is a coma-like sleep by day in the classic TV show. Granted Dracula, too, falls into a similar sleep but he can still move about by day if he chooses not to sleep.
2. Barnabas in the TV series could take the form of a bat (and Depp's version doesn't shapeshift at all, at least not in any footage kept for the finished film). Dracula, however, can take the form of a wolf, bat, or mist by night.
3. Both Barnabas and Dracula know a great deal about the occult and in the TV series Barnabas could cast a few spells (though this was plot point used for convenience and often forgotten) and Dracula, in Stoker's novel, has studied at Schomance (a school of magick in Eastern European folklore).
4. Barnabas was made into a vampire by witch, Angelique, Stoker's novel never says how Dracula became a vampire and it seems he, himself, might not know as he appears to have simply woken up after death as a nosferatu (vampire). This gives Dracula a bit more freedom with his origin.
5. Though Barnabas and Dracula have a psychic link with their victims, Dracula can transmit actual thoughts to them.
6. Age: Dracula, if he was real would be 581 years old (the historical figure was born in 1431). Barnabas, however, is just barely over two hundred and fifty years old. .
7. Dracula has a castle, houses in London and Carfax in Whitlby. Barnabas has... The old House on the Collins estate or if you go by the new film, a guest room in Collinwood.
8. Count Dracula used to be Prince of Wallachia. Barnabas used to be the son of a wealthy business man ir if you go buy the new film "Lord of Collinwood manor" in rural Maine. This gives Dracula the superior title.
9. Count Dracula is also a knight, a member of The Order of the Dragon. Barnabas is... A member of the Collins family.
10. Dracula has had over two hundred film incarnations, animes, manga, comic books, TV series and books including a book series from his point of view by the late Fred Saberhagen. Barnabas however has only had two TV shows, one failed pilot, two movies (not including Night of Dark Shadows as he wasn't in that), some comic books,a nd some novels.
11. Dracula is based on a very violent and psychologically scarred historical figure with an incinlation for sadism toward his enemies. Barnabas, however, is based on Dracula.
12. Dracula can summon wolves to do his bidding and can control the weather. Barnabas makes dogs howl when he's going to do something vicious (original show, not the Depp movie).
13. This one is kind of a cheat but if you go by the novel An old Friend of the Family by Fred Saberhagen both Count Dracula and Barnabas have dysfunctional descendants. However in An old Friend of the family (also set in the seventies) Dracula tracks down the ones accountable and makes them pay, brutally. Though Barnabas has done similar (such as the torture and slow death of Trask in the TV series) Dracula resolves a lot more a lot faster than Barnabas.
I love Barnabas but Dracula is simply the superior vampire.
Oh, and to the one who listed Nosferatu. Nosferatu is not actually his name. It's his race. Nosferatu (an old word for vampire) means "Not dead." the character in Nosferatu is Count Orlock or if you go by Shadow of the vampire than the actor Max Shrek himself is really a vampire.