Library schools have forever been about school, academic, public, and special libraries (law, medical, corporate, and others). They've also been about archives and records management. When library schools started to get more into training other sorts of "information professionals," they broadened their names to include various sorts of "information." More recently, there has been the iSchool movement which defines itself more broadly. iSchools include some schools that do not even offer an ALA accredited MLS (or MLIS or MSLIS, etc) - the traditional librarian degree either because they dropped it or because they never had it. Even iSchools that offer the MLS, like Maryland (affiliated but not speaking for them), have branched out to new degrees that require some of the same libr* skills but are not for librarians and are not ALA accredited. They are simply different degrees. Maryland - just used as an example because I know it - offers a MIM (master of information management) and an HCIM (master of human-computer interaction). The MIM graduate at work is an information architect and head of the usability team.
Well. Many people know about ALA because it's so massive and so political. Fewer people know about ASIST. ASIST has always drawn information people who are not strictly dealing with libraries. Information retrieval, bibliometrics, visualization, and other areas. It's more library school and now iSchool researchers and many fewer practitioners although from time to time they do make a big push to be more welcoming to practitioners (practical application papers are not accepted very often).
Most professional librarian jobs require an ALA accredited MLS. The accreditation committee has increasingly shown concern over the move away from libraries by the iSchools. For example, in some schools, all of the routine how-to-be-an-actual-librarian classes are now taught by adjuncts because there aren't any qualified faculty. A class on cataloging, for example. Pretty important! The ALA accredited degree is specifically to show that the school meets the requirements from a professional library association - for librarians, who often work in libraries doing library stuff.
ASIST, has come out with a statement that is basically like hey, lots of our members have nothing to do with libraries and ALA is being mean to require in their accreditation that the school do all sorts of library stuff. Here's a quote.
It is imperative that accreditation standards be comprehensive and flexible enough to accurately represent educational requirements in multiple information fields, both in and outside of libraries, archives, and other longstanding information organizations. Accreditation must reflect the eclectic, diverse and pluralistic nature of the information field and must be fully applicable to an array of information professions. As a result, we call for the ongoing dialog between ALA and ASIS&T on accreditation issues.
Dorothea Salo (erstwhile blogger here at Scientopia), a faculty member at an iSchool supports this although she is not fond of ASIST (as I am, by the way). I believe her thinking is of the lines that library school is more about approaches, broad skills, and professionalization and can't and shouldn't teach all the nitpicky details that vary from place to place and change quickly over time. (like you didn't teach me to program in Python even though I went to library school before it was invented! you didn't teach me how to install this catalog software!).
With all of that said, I say to ASIST buzz off and butt out. Seriously. If they want to do their own accreditation (when they can't even agree on what we do or who we are), fine. ALA has no obligation to really include them at all. It makes sense that ALA at least talk to ALISE members - those are the library science educators - to coordinate big changes. Probably ASIST members who care about this are in ALISE anyway.