indyrank - I can see how the second paragraph of my answer might have been the cause of some confusion. To add more nuance to my answer...
First, the question I answered, with my emphasis added, is...
...does the on-page SEO benefit of site A get transferred in a 301 redirect or does only the link juice get transferred?
Again... there is no longer any onpage benefit to be transferred, because site A no longer exists.
I also said: "Only the link juice gets transferred"
. As aakk9999 points out, while "anchor text benefits" get "transferred", all sorts of other factors related to the link source and destination enter into the effects of that link. It's not only about anchor text.
To clarify further, though, I'm saying in my second paragraph that only the anchor text benefits coming into A
get transferred to B. That's because, in a 301 A->B redirect, B is replacing A. This needs to be considered on a page-by-page (or, more precisely, on a url-by-url) basis.
What I'm excluding from the transfer are any effects of internal anchor text and link juice redistribution that had occurred within the site A. If you think about it, site A's internal nav structure and offsite linking are onpage factors, and they disappear in the redirection of A to B. As I put it...
...the outbound link juice from links that were on the page disappears as well.
This is true whether the redirects are done on a sitewide basis, or on a "page-by-page / url-by-url" basis. The internal structure of site A is lost in the redirect even if each page is redirected individually.
That said, if, eg, only the domain name were changed, and if A and B were otherwise identical, then a "page-by-page / url-by-url" redirect would apparently preserve the same link juice flow, but it actually would be the reproduction of internal nav and site structure that would be doing the preservation.