Press can be made accountable for what it reports to a large extent, outright lies can be called out and the newspaper or whoever signs the byline can be sued in extreme cases.
An anonymous fake news article has no risk or accountability for its author, that's the main difference.
With press, at least everyone is reading the same lies. Hyper-targeted fake news makes it impossible to discuss anything, because across groups, everyone has their own set of "facts", making people talk past each other.
For any given news story, there are usually few "traditional" sources covering it, with only a fraction of them being widely popular. It's a manageable amount. Social media has switched the way people consume news (and content in general) - they're no longer consumed on per-source basis, but on per-story basis. I.e. most people don't read a particular news site, they read whatever stories from whatever sources happen to be on their Reddit or Facebook feed.
An anonymous fake news article has no risk or accountability for its author, that's the main difference.