Difference between Don't color manage and use working color profile

Super quick and dirty: Don't Color Manage is 99 times out of 100, a bad idea.

Old, but still the same over the years, quick read:

http://digitaldog.net/files/06AssignProfileCommand.pdf

Newer, longer video and more details:

 http://digitaldog.net/files/PhotoshopColorSettings.mp4

Photoshop CC Color Settings and Assign/Convert to Profile video

Don't colour manage removes the current profile and or sets an explicit flag to not embed an ICC profile.

https://helpx.adobe.com/photoshop/using/working-with-color-profiles.html

It means you're in trouble either way 😉 You don't want any of those.

Any document should always have an embedded color profile. This profile should follow the file wherever it goes - and if it's embedded, it will. An embedded profile will always override your color settings.

You normally don't need to assign a profile, because it should already be there. However, if for some reason there is no profile, then you need to assign one. But this isn't arbitrary. You should assign the profile that corresponds to the color space the file was actually created in. In this case, the working space matters, because lacking a profile, the image is presented in your working space. So that's the profile you need to assign (unless that looks positively wrong).

Bottom line, always make sure there is an embedded profile, and if there isn't, assign the correct one. The one thing you should never do, is "don't color manage this document". Don't even think about it.