To help understand, let's draw a gradient, a big red on the left and a gray on the right, and then use a blue brush (#00c0ff) to draw one in hue and color mode respectively.
After modifying the gradient with hue mode, keep the original saturation change, with blue on the left and gray with low saturation on the right. The "color" mixing mode changes hue and saturation at the same time, and the original gray also turns blue.
Of course, you can also try a very dull color (#a7ac9c, a little green and gray):
In Tone mode, the gradient keeps the original saturation change. Although the color of the brush is not bright, it can be seen that the big red part has turned into bright green. In the "color" mode, the whole painting is the same gray-green.