I think OCTGN got closer to the tabletop experience, where you btw also have to shuffle the deck yourself before drawing a setup etc.
Also, it lets you forget things, which at least for me has always pretty much been the main thing to do while playing AGoT: Remember all the effects that are currently active, remember all the responses, remember all the cards your opponent could have, remember all the cards you put in your deck and could draw into ...
In a physical game there is no popup asking you to trigger Put to the Sword after you won a mil challenge by 5 and have 2 gold left over. You have to think of that yourself.
And while that might sound mundane, pretty much all games I lost back when I was actively playing were because of me forgetting things.
So while theironthrone.net certainly makes it more accessible to a large group of casual players, as you put it, I think it waters down the experience.
Updating was a pain in the ass.
It not running on Mac I refuse to believe, since things like Wine exist.