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The answer to this question will involve a long discussion regarding color management. The short version is that with color management enabled (the default behavior… and the one you should be using), Blender uses a linear color space internally. This means that the colors you get from web references (usually RGB values ranging from 0 to 255) will not be accurate if you convert them to the normalized (0.0 to 1.0) values in Blender’s RGB sliders. However, the exception to this is the Hex input. The “Gamma Corrected” text is there to let you know that the hex value you’re inputting is not on a linear scale and therefore should exactly match the colors cited in a web reference.
So, the shorter version is that unless you’ve disabled color management for some reason, the color you get from inputting a hex value should be exactly the same as you’re getting from your web references. If there’s a difference, you should report a bug and include an example .blend file that shows the issue in question.