When you save an image in Photoshop, you can choose to embed the profile in the image data. The embedded profile will be what instructs color-aware applications in terms of rendering the colors correctly, according to the embedded color space. So, the embedded profile tells the color-aware application how to interpret the RGB values in the file - how those actually get displayed on YOUR display is a function of your display calibration and profile. But, at least, an embedded profile will tell everyone's system to display the RGB numbers according to, for example, sRGB. Your specific display calibration and profile can make those values look different, potentially, than someone else's if your display (or theirs) is set for different display characteristics or has a bad display profile.
For example, if the embedded sRGB profile told all color aware browsers to display 180R, 130G, 77B as Candy Apple Red, all color-aware applications would know to display Candy Apple Red, as opposed to Fire Engine Red. However, even if your color-aware browser knows to display Candy Apple Red, your version on your display may look different than Joe's version on Joe's display because Joe has his display calibrated and profiled to different white point target, luminance, etc.
If you do not embed any profile, then roll the dice. I suppose most applications and systems will interpret the RGB numbers close to something like sRGB - so in this respect, the numbers will be rendered the same, regardless of the color space in which they were created or meant to be seen. But hypothetically, one application might interpret 180R, 130G, 77B as Candy Apple Red, while another might interpret it as Fire Engine Red, and another might interpret it as Fire Ball Red - all on the same system. This is a problem, dramatized here for effect, but you get the idea.
This is commonly seen when someone posts an image here in AdobeRGB with an embedded profile, but viewed on non-color-aware browsers. The color-aware browsers know how to interpret the RGB numbers and display the image colors as intended by the embedded profile instructions - non-color aware browsers do not and assume the color model, often times incorrectly. If the same image were posted with RGB numbers intended for Adobe RGB and NO profile was embedded, this would also cause non-intended changes, but likely more uniform for all viewers because there would not be such an explicit difference between the browsers that know what the artist wanted (the color ware browsers that can honor an embedded profile) and the browsers that do not know what the artist wanted.
I know even less about web color, so I will leave commenting on the mysterious "save for web" shift.
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