It's good that Google learned from this problem, and built a solution into their naming when releasing Chrome; the open source project is named Chromium, and is what distro versions can be named, and the official build from Google is named Chrome.
It would be nice if Mozilla would follow suit, and come up with an official name for the open source project and community builds which is distinct from their official branded builds. Right now Debian uses "IceWeasel", GNU uses "IceCat", and other distros would have to come up with their own name if they made changes that were incompatible with Mozilla's trademark policy.
On the other hand, in pretty much every repository, you find "Chromium", but if you instead download straight from Google you get "Chrome".
Chromium and Chrome differ in a number of ways beyond the name and trademark, though.
Iceweasel is literally just Firefox packaged for Debian, except with a different name and icon. I don't think there is any functional difference (beyond the normal distro-specific packaging that every distro does).
Chromium lacks a number of features that Chrome has, because Chrome comes bundled with a bunch of proprietary components that will never be distributed with Chromium.
A basic example is that Chromium cannot (without a patch) display PDFs in-browser, the way Chrome can. Chromium also cannot support Chromecast, or any of the many other Chrome-specific features, of which there are many[0]
[0] https://i.imgur.com/AIxYzl9.jpg (screenshot is from Firefox, but it illustrates how much Google focuses on Chrome-specific functionality.)
Iceweasel is literally just Firefox packaged for Debian,
except with a different name and icon.
If you read this bug report[1], it's really about more than just a "different name and icon." If they used the official name and icon, they would have needed to have every patch be reviewed by Mozilla, which has serious problems for urgent security patches. And in that discussion, Mozilla voiced concern about several of Debian's patches.
It looks like there has been some effort to do what I've described, come up with a generic default name for Firefox builds that can be used as a default, unbranded version[2] rather than requiring every distro that doesn't want to or can't comply with Mozilla's trademark policy have to do it on their own, but the effort appears to have stalled.
Chromium lacks a number of features that Chrome has,
because Chrome comes bundled with a bunch of proprietary
components that will never be distributed with Chromium.
Sure, I'm not saying that the situations are exactly analogous, just that Chromium has managed to much better avoid the kind of trademark issues with distros that caused so much trouble for Firefox and Debian. On the other hand, as pointed out elsewhere in the thread, it appears that there is no official Chromium trademark policy (at least easily discoverable), but it seems that most distros have been able to use the name without complaint from Google.
It's somewhat trivial, but I remember one change that precipitated Mozilla's complaint was a patch that stopped GIF animation when hitting the escape key. This was years ago though, so I don't know offhand what functionality Debian currently maintains that isn't in mainline Firefox.
Yeah, but IceCat is not the same as IceWeasel (it has different patches), so it makes sense for them to have different names.
Mozilla lets you use Firefox if you don't change the source; I believe Fedora and other distros release it with the original name. The difference is that, unlike Debian, they don't apply patches to the upstream source in their builds.
And I find it hard to believe that Google allows you to use the Chromium mark if you release a patched build; they're probably no different than Mozilla in that regard.
It would be nice if Mozilla would follow suit, and come up with an official name for the open source project and community builds which is distinct from their official branded builds. Right now Debian uses "IceWeasel", GNU uses "IceCat", and other distros would have to come up with their own name if they made changes that were incompatible with Mozilla's trademark policy.
On the other hand, in pretty much every repository, you find "Chromium", but if you instead download straight from Google you get "Chrome".