I think, for me, the difference lies between constructive comments (which I welcome, even if they disagree) vs hateful/spiteful comments (which I would delete but gratefully haven't received to date).
On my fear of flying piece, I was touched so many great comments were from people not afraid to fly...yet they read my article. If someone said, "you idiot, I don't get wimps who are afraid to fly" that adds nothing to my article. If someone listed all of the benefits of flying I am missing out on, how could that not add richness and well-roundedness to my article? That is how I do it...would love to hear how others decide what to post or delete.
Though I hold comments to scan for spammy posts, I don't want to censor those who don't agree with me and I think it adds something for my readers, so I post them no matter their view. There are all sorts of writers out there - that's just the type I am - I have my opinion pieces but love to engage in discussion and hear something new...feeling certain that learning is a journey without an end and unwilling to ever think I know all I need to in this life.