Embed, tag, Assign are all terms used that describe associating an ICC profile to an image. Without the profile, the numbers have no associated scale**. The profile tells ICC aware applications where the numbers lie within human vision. Without, Photoshop and other ICC aware applications don't know the scale of the numbers so they guess at how to preview those numbers. This is why when you assign a profile, the preview changes but the numbers do not.
There's a tutorial on Convert versus Assign on my web site should working thorough the various processes make more sense.
** case in point. I'm seeing the wind and snow blowing outside my window and my weather center tells me the wind is blowing at 10. 10 what? Knots? MPH? Without the scale, the number isn't very useful.