First you'd have to better define what do you mean by "your operating system"? The one on your development machine or one on the target machines?
As for productivity, I'd say there isn't huge influence by operating system on my development machine. A lot of the tools you're going to use are either entirely cross platform (for example all Eclipse based IDEs), or at least the core of it is cross platform (revision control systems like SVN, Git or Hg). And even for the things that are not cross platform, it's more about tooling than OS tricks.
It's a other story if we're talking about target OS. You do need to know it, although I wouldn't say it's about tricks or hidden features. Just good, general knowledge. Like for example working with Unices it's very useful to know shell commands, know basic tools like
sed, know the directory structure, knowing the permissions system, knowing what symlinks and hardlinks are. But I wouldn't say, that average developer needs to know for example about inode structure or kernel internals. On the other hand being used to developing on Linux I've had bit of a struggle with Windows (many things you take for granted on Unices don't work on Windows, like for example file locks are completely different).
On putting that on CV -- depends how it fits in with your other experience. If you're just starting, than it might be something worth mentioning, otherwise it might be implied by your other experience. Personally I don't put knowledge of OS on CV, I do however put knowledge of programming OS internals there.