In many ways Academia.edu and ResearchGate also function as a digital CV, listing research experience and different forms of publications, whether it pertains to academic papers or other forms of publications, such as conference papers or blogs. But if that's true, why would an academic bother setting up a LinkedIn profile?
Difference in audience
The answer would largely lie in the fact that academic social media networks have a different audience than LinkedIn. ResearchGate or Academia.edu are interesting to build a (collaborative) network of academic peers, while LinkedIn allows networking on a broader scale, also reaching out to the private sector and civil services. And of course it doesn't have to be either/or. In fact, these can most definitely be used as complementary profiles.
Value is in the difference
For example, if you compare Dr. Christian Taube's ResearchGate profile to his LinkedIn profile you'll notice that while both of them tell you he's currently Head of Pulmonology at the LUMC, it's only at the ResearchGate one where you'll find what his research is focused on and an overview of his publications. Similarly for Thijs Porck's Academia.edu profile and his LinkedIn profile; on both you'll be able to discover that he's a PhD-student and teacher at Leiden University and that his field of interest is Old English philology. Yet on his Academia.edu profile you can not only find an overview of his publications, but also an overview of conference papers, presentations and blog posts, which are lacking on his LinkedIn page. On the other hand, his LinkedIn page shows his research and teaching experience and also the breadth of his other activities, such as editing a scholarly journal and various administrative functions within his department. If all of this information were to be placed in one profile, it would become cluttered and make finding the information you were after more difficult.
Profile, network, promote
If it were truly a contest between ResearchGate and Academia.edu versus LinkedIn, then it would end in a draw. All of them have their own focus and audience and if you want to reach a broad spectrum of people and have all your pertinent information easily available to peers, funders, and the public then having multiple profiles to network on would seem the surest course. And remember to let people know you are there, so they can find you!