LinkedIn will replace the traditional academic CV. My short answer to that is "no". This isn't because I think LinkedIn is bad (it's not), or that the CV is awesome (it's not). I've got a bone to pick with the traditional, paper-based, academic CV.
The common wisdom, as Kim alludes to, is that a resume is short and targeted, while a CV is longer and is meant to include everything (and the kitchen sink) in your career. Resumes, for me, seem constraining. How can you adequately describe yourself in 2 pages, especially for seasoned professionals who are older than I am and have a wealth of knowledge and skills? At the same time a resume is a creative puzzle to solve. It's a tool for communicating what you will bring to the team you want to be hired into when you apply for a job. A resume encourages your to look into a company and a department, and tailor it to fit where you want to be. It fits a narrative.
By comparison, the academic CV is lazy. It's just a list of everything, and you hope that it's well formatted. The onus is on the reader to go through the (reams of paper that comprise the) CV and make sense of it. Despite the CV being the lazy approach, I also think that the CV is also nothing more than a library index for your career. It's really just the headlines for what you've done, but it really lacks substance. For example, let's say I am a member of 5 professional associations. Great. There is a sub-heading on there for that. But what do I do in those? Or, let's say that I have a section for conference presentations. Great. What was the presentation all about? Is the presentation available on SlideShare (or on your website, if you want to keep it on your own domain)? Or is the recording of the presentation somewhere so I can listen to it? Ditto for academic papers published. Now, of course, I could go out and find those and read them (I am privileged enough to have access to library databases), but that is detective work on my part, as reader of a CV, to do. Got grants? Great! What did you do in them? Who cares how much money you got? (well, I suppose a chancellor, provost, or dean might care...) - the important part is what was the impact and what did you learn? It doesn't matter if you got a million-dollar-big-whoop-grant if you didn't use it effectively and nothing came of it.
Don't get me wrong. I like to keep a private CV for myself. It is a nice reminder of the work I've done, the papers I've published, the awesome collaborators I've worked with over the years, but that's just a mental index for me. It could be meaningless to other people. It seems to me that CVs are more about shock and awe, a "my CV is lengthier than your CV, hence I am better". I find this a bit of an irony because one of the critiques I've heard from academics about LinkedIn is that the approach that LinkedIn takes is very quantifiable (in that you list, and list, and list) and it dehumanizes the person, but I see CVs being used in the same way.
I have a solution to this - but it's not pretty --> some form of linked data. LinkedIn works well for non-academic jobs. Sure, they've added ways of putting in your publications, but you end up having a scroll-a-thon of infinite screens whenever academics start adding in their publications. So, keep linkedin for your non-academic jobs. Use Publons to keep a record of your peer reviews. Use Google Scholar to keep track of your published work, and have that work linked so that people can easily find it (publish open access and/or self-archive). Use other services to keep track other relevant accomplishments. The problem that I see is the third-party-ness of these services. LinkedIn might be killed off by Microsoft. Publons might not turn out to be profitable and it's gone. Academia.edu might decide to merge with ResearchGate - whatever it is, the point is that you don't have control of these services, and that's a problem.
I wish there were some sort of WordPress-style software that you could host on your own servers, keep your data, but at the same time be part of a global network that connects your work to others, so you could see who else was in my department at the time I served there, or what my co-authors went on to do after our collaborations ended, or what papers cite the work that my co-authors and I produced. A single system, disconnected, just keeps the CV in another form. From paper to some sort of digital. What we need is an evolution of the CV.