Dithering is sort of like having the ability to "blend" any two colors of your palette (possibly even more than any two, if you use it well), so instead of being a 16-color pallete, it's like working with a 16+15+14+13+12+...=136-color pallete. It's a drastic difference (at the cost of graininess, of course).

Tried this online tool https://onlinetools.com/image/apply-dithering-to-image and Floyd and Atkinson both look great, Atkinson a bit better.

Dithering is still more important than is commonly known, even with 24-bit "true color". For example, imagine that you had a gradient that goes from white to black across a 1920x1080 monitor. 24-bit color means you only have 256 levels of color, so a naive gradient implementation will result in 256 discrete bands of different grays, each about 8 pixels wide (about as wide as this "w" character).

You might not thing that you'd notice that, but it looks surprisingly bad. Your eyes would immediately notice that there are "stripes" of solid gray instead of a smooth continuum. But if you apply dithering, your eyes won't be able to notice (at least not easily). It will all look smooth again.

In a situation like this, I like to use "blue noise" dithering, but there are scores of dithering methods to choose from.