Does the manga contain more story than the anime?
To be exact, the anime deviated from the manga slightly near to the end as instead of Zero and Yuki being on a roof, Zero is standing on a platform made entirely of the thorn vines from Bloody Rose. Also, Kaname killed all the Vampire Senate, not just Ichijo and the guards at the very end. After that, there is a time skip until Cross Academy is rebuilt and running again and Kaname is now living with Yuki.
Pretty much from the small discrepancies at the end of the anime, you can start reading from Chapter 49 (Volume 11 - 49th Night) and not really miss out, though I would just start reading from chapter 1 again to read the Bonus/Extra stories that most volumes came with (I don't know how much they add, but I seem to recall a small story of Kaien caring for little Yuki)
Since the series continues on after Yuki leaves Cross Academy with Kaname, there is more information as we learn more about Purebloods and what happened between Kaname's Parents and Rido, particularly why in the anime Juri says that Rido "already killed one of her children" when Kaname never mentions another sibling apart from Yuki.
We also learn more about the Vampire Hunters and why their weapons are so effective against Vampires, and finally learn that Kaname's real end goal is for his past actions (killing Shizuka Hio, wiping out the senate, making Zero what he is)
Does it finish the story off?
Yes, and according to this answer, the manga finished in 2013 in Japan with the English translation done the next year. It's probably safe to assume that after volume 19 (Chapter 93), that's it.
Of course, whether or not you accept that it's the end is another matter. Since there have been no new releases afterward, there doesn't seem to be any plans to continue after Chapter 93. There's probably fan fiction and doujinshi floating around, but these wouldn't be considered as canon (e.g. Yuki making Sayori a vampire and becoming a couple forever)