Here's my take on why the double standard exists: yes, both Haley and Nathan made some serious mistakes, but Haley had a LOT farther to fall.
Since day one, Haley has been shown to be responsible, mature, level-headed, etc. She also came across as a bit superior, sanctimonious and holier-than-thou at times. So I believe that a lot of people feel some level of joy in seeing someone who is portrayed as trying to be perfect seriously screw up.
Nathan, by contrast, well we all know where he came from. And he's come a LONG way from there, but he's still going to mess up because he's always going to be fighting against that guy he used to be. And sometimes he's gonna lose and people accept that.