Questions keep coming up as I want to achieve a good browser/OS coverage:
Minor browser version change: For instance, most people use Chrome 56, 57 and 58, then do I need to do end-to-end test against each of them?
OS version change: similar topic, for a given browser version, do I need to test against Windows 7, Windows 8 and Windows 10 separately?
To hopefully answer your questions:
If you're building web applications, I wouldn't worry too much about a browser incrementing to a new version. If you are using experimental features of a browser (meaning features that aren't yet complete) then you might have to, just because some things change during the development of those tools and API's and can impose breaking changes for your app. I would recommend reading about changes that are made with new versions anyways just to see if anything spikes your interest.
I'm not a software developer, but I would assume the same principles apply as stated above with the caveat that OS versions and browser versions are in different ballparks. You may need to write extra code to support new versions of OS updates that are a big version change (i.e. Windows 7 => Windows 8 => Windows 10, etc.)