The difference between these services is that PlayStation Now and, presumably, xCloud, will be streaming games you don't own via a subscription service model whereas Valve is simply making your own games available to you for free - a model that will appeal to more cost-sensitive gamers and those of us with already-massive Steam libraries.
Previous versions of Steam Link allowed players to connect to a machine running Steam from a Steam Link or Steam Link app, but that was only possible so long as both devices were running on the same wifi network.More news: Trump Does Not See White Nationalism As A Rising Threat
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Steam Link was originally released as a hardware device in 2015, but Valve later discontinued it in favor of an app for mobile devices and smart TVs.
Today, Valve announced Steam Link Anywhere, which allows users to play their Steam games from anywhere with a decent connection using the Steam Link app or physical Steam Link hardware. Steam Link Anywhere is a new service and it greatly expands upon the previous functionality of Steam Link for Android. You'll have to opt into the Steam client beta with the date of March 13 or newer, and then you can connect to any computer by selecting "Other Computer" from the search.
Valve's announcement of Steam Link Anywhere was pretty short and sweet. This was only really useful for those times when you want to play a game on your couch. Part of this is better development tools, but more importantly, Valve are letting developers route network traffic through Steam's own infrastructure now.