This refers to settings on how to treat different settings in linked images. Illustrator doesn't remember what was chosen when each image was placed into the file. So you have to re-select it every time you open the file. You might not know what someone has chosen who handed a file to you. So Illustrator at least informs you that someone at least had a different default setting than you have, thereby assuming that people will always want to treat each and every image the way they set up their color management.
Yes, it is a vital setting, because it determines how colors will be converted when embedding the file, printing or savng a PDF.
I am a native English speaker but STILL do not understand what this dialogue box means, even after reading your explanation. More importantly, though, since you indicate that this is indeed a "vital" setting, I do not yet know what to DO about it when I see it.
I created a document in Adobe Illustrator CC 17.0, and then installed CC2014 some time later— it is when I try to open the document in this new version of Illustrator that I get this same error message. I take it that the message means there is some kind of problem, though the file appears to open normally. Is there something specific I should do to the image/ file in order to prevent Illustrator from showing me this error message? What happens if I routinely ignore the message? Thanks!
Most imporant: set up color management. Synchronize color management across your applications and versions.
If you don't know how to do that, you're in really serious trouble. There's some literature on the topic. Get it and read it. You might start with this:
and the manual. Look for a Lynda.com training or search the web for resources. There's some really good stuff available in German, but I don't know about the English resources (beacause our own are so luxurious).
The dialog box tells you that your settings regarding linked CMYK images were different in your other Illustrator version. You had it set to "Preserve embedded profiles", but now you've set it to "Preserve numbers". Depending on the nature of your files, your workflow, your co-workers' workflows or the printing process both settings might or might not be correct. But most probably changing the settings in the middle of working at the same file is not correct. The setting influences how colors are converted when creating the final output. So, yes, it is important.