Tab Groups are cool, but I don't think anyone uses them. If you're on Windows 7 it hides the menu bar for more vertical space. It has support for lightweight extensions (called "JetPacks" internally) which don't require a restart to install or uninstall. There's no statusbar, like Chrome, URLs pop up when you hover over a hyperlink.
The main advantage over Chrome is that it doesn't freeze up when it's loading a new tab.
I use Tab Groups mainly for task related stuff, as well as using a tab group to allow for a read later style of interface. But no, it's not a major thing for me, I've used Chrome for the past few months and not missed Tab groups.
Tab groups would've been much better if they showed all the tabs across all windows (like Expose on Mac does for windows). Unfortunately it's just a gimmick in its current implementation.
Firefox also hides the menubar on linux, although it can also be disabled. It is also posiblse to enable an 'extension bar', which takes the place of the status bar for extension icons.
The main advantage over Chrome is that it doesn't freeze up when it's loading a new tab.