But without knowing any better. Chrome has always updated on its own and there's no reason to know what version of Chrome you're using because it's the latest. I had to check just now, apparently it's 12. FireFox has always had hugely publicized major updates with a long time between them, so there's a disconnect--you're supposed to care about a new FF version.
Chrome updates also result in JavaScript breakage.
File/Blob API is an example that, while being experimental and a moving target, happened on the background and broke from one launch of the browser to the other without the dev knowing what the heck was going on.
Same breakage happened from Firefox 4 to 5, but at least the end user (the dev in this case) knows that something important happened and might be related to that. This is key.