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I have no idea why Microsoft chose this name, nor why the branding geniuses in Redmond thought this was a way to dispel confusion, but live with it we must. If you think of Skype for Business Server as Lync Server 2015, you will immediately see all of this lock into place.
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Just forget about the word Lync, and you will pretty well have it right. Lync is now Skype for Business, thus making all on-premises clients and servers Skype for Business clients and servers. Some people have gotten the impression (and I can't blame them, because the naming is confusing and Microsoft has done a poor job of communicating exactly what this release is) that Skype for Business applies only to the Office 365 service, and on-premises deployments will continue to run Lync. Skype for Business is not a cloud-only service. But the guts of the service - the components, configurability, administrative aspects and concepts - everything that applied to Lync Server 2010 and Lync Server 2013 (as well as to Lync Online, the cloud version that is a component of Office 365) still applies in almost exactly the same way to Skype for Business. It is a rebranding and a change to the user interface, mainly to "blue" it up and make it look a little bit more in line with the consumer Skype client (albeit without the advertisements pervasive in the free version). Skype for Business is not a reinvention of the Lync service.
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It operates independently of Lync/Skype for Business, although some companies have enabled federation, which is the ability for Lync clients to initiate chats to Skype users.
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Skype that you freely download is still Skype, the same customer service Microsoft purchased from eBay a few years ago. Yes, it seems Microsoft has done it again - it has created two services that seemingly have the same purpose and outcome but go about it in entirely different and mostly incompatible ways (think OneDrive and OneDrive for Business). Skype for Business is not a replacement for Skype.
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