HackerNews is comment driven, but does a decent job of facilitating discussions - but not particularly deeply. Reddit is similar. Forums are much more amenable to linear, deep, discussion between a few parties, but can also facilitate comments. Both have their place on the internet, and I don't think that forums are necessarily the answer to everything, but it feels like a lot of people left those communities to end up in Reddit and that's a shame.
If you reply to an earlier one, the UI will handle it and show a new branch where that conversation started. I'm not a huge fan of typical mailing list web UX and usually view flat, but it's so common that I guess a lot of people like it. My opinion is probably biased from years of using/running forums, so I'm much more used to that UI.
I think newer systems like Discourse will formally track replies to you (versus a new message in the overall thread), while legacy forums like PHPBB usually just quote-reply.