A set of 216 colour values is commonly considered to be the "web-safe" colour palette,
developed at a time when many computer displays were only capable of displaying 256
colours.
A set of colours was needed that could be shown without dithering on 256-colour displays.
The browser-safe palette was developed by programmers with no design sense. That's
because a designer would have never picked these colours. Mostly, the palette contains
far less light and dark colours and is heavy on highly saturated colours and low
on muted, tinted or toned colours.
The only reason to use the browser-safe palette is if you have a concern that your
Web design work will be viewed from a 256 colour (8-bit) computer system.
Today, the minority have them, so the justification for using the browser-safe palette
has diminished greatly if you are developing your site for users who have current
computer systems.
Conversely, there's no harm in using the browser-safe palette either. It simply limits
your choices to 216 colours. Most people don't have a lot of colour picking confidence,
and working with limited colour choices is easier.
At this point, there's no right or wrong when it comes to which colours you pick,
but more important to know how to combine colours in pleasing and effective ways.