"Weighted" keeps the perceived luminance of the original color intact. So if your original clip is blue, and you're colorizing it to red, you will have a more muted red as a result. If your original clip is yellow, you will have a more intense red. This is because the perceived luminance of yellow is higher than that of blue.
If you toggle Weighted off, the result will just be a simple multiplication of the RGB values of the color you pick, with the actual luminance or greyscale value of the source clip. So black will remain black, grey will become a percentage of the colorize color, white will become 100% the colorize color.
"Palette" is a simple distribution of the palette by luminance of the source pixel. So where the source image is black it will show the swatch on the left of the palette, where the source image is white, it will show the swatch on the right of the palette. The swatches in between are distributed linearly depending on the greyscale value of the image.
Personally, I'm a big fan of the unweighted option.