Tabs now appear in the recent apps list (which can be shut off and turn back into normal tabs in Chrome), and it seems to be tied heavily into Google Now (though I'm not sure which parts are which).
Chrome on a Nexus device is a lot like Firefox on a FirefoxOS device, if that makes any sense.
I also have a Nexus 6 device and personally I don't find the integration with the recent apps list useful. Unless we are talking about apps, otherwise for browsing I much prefer to have those tabs grouped together in one app. Plus Chrome's "normal" tabs where more annoying to use than Firefox (continuing the tradition from the desktop).
The thing that bothers me about Firefox is the broken behavior of "Add to Home Screen", which doesn't listen to the website's Manifest.json. There are issues raised in Firefox's issue tracker for this and this will get fixed, but currently if you want to pretend that a website is an application, then Chrome is better.
I'm doing that for a small utility I've built for myself: https://parolamea.org/manifest.json ... funny thing is that even though Chrome is sexier to use for this use-case, I'm still using that website in Firefox because Chrome doesn't offer to save passwords for "ajax logins", a known limitation.
But either way, I end up using both. I'm using Chrome for apps (e.g. I prefer to access Facebook and Twitter through Chrome, as it's better for privacy and I don't want notifications and all that crap). I'm using Firefox for browsing, as the default.
Here are some reasons for why I like it ... it has the Awesome Bar (e.g. much better search capabilities in your own history), it has the Reader View + Reading List (i.e. I don't need Pocket or Readability, although if you like Pocket, then it has the best integration available), it has AdBlock Plus, uBlock and an alpha but functional version of NoScript and really, many websites are simply unusable on mobile without an ad blocker.
On Chromecast support, I don't know about the desktop, but if it's not supported on Firefox desktop already, then it will probably be supported pretty soon. Other mobile features have leaked into the desktop version as well, for example Reader View.
Ah, none of the tab stuff or app-saving from Chrome are features I would use. And I believe my phone just supports Chromecast screen sharing at the OS level (alongside DLNA and Miracast). Is that an addition from Sony that isn't in stock Android?
EDIT: I see that Chrome is "deeply integrated" with my phone in the sense that it can't be uninstalled, but I'm sure that's not what you mean...