With the new Custom Colors option for Custom Design, should we remove the color changes that we've done through CSS coding, and instead do it through this?
It's possible to use both, but I would recommend removing current color-related CSS and then trying out Custom Colors so you can see what it's really doing. Once you find something that you like, you can add back CSS code to make further adjustments if needed.
Is there a benefit to one over the other?
The custom colors option was designed to be crazy simple to use, but it's also designed to work on a a set of elements as defined by a custom color mapping that is specific to each theme and made by one of our theme team developers. That means some of the choice surrounding which colors change with Custom Colors is determined by the theme. Custom Colors is probably the best choice for anyone who doesn't want to mess with any CSS code, and the CSS Editor is best for making fine-grained, super-specific color updates using CSS code.