It's a bit like 2 space indention versus 4 or 8 space. I find the em-dash example pretty compelling because visually relaxes my eye more than commas do — similar to how prefer at least 4 space indention over 2.
(Unrelated: The alt-code for em-dash on Windows is Alt+0151. I just typed my first manual em-dash wow exciting.)
As far as I know, the UK styles that recommend spaced dashes are using en dashes where US English would use em dashes, and the ones that actually use em dashes, like Cambridge or Oxford, are unspaced.
(Unrelated: The alt-code for em-dash on Windows is Alt+0151. I just typed my first manual em-dash wow exciting.)