As Billy Kerr pointed out in a comment, a reason sprites of the same size and resolution look better in modern pixel art than in classic pixel art is that a modern desktop computer generally has at least a 16-bit colour rendering ability. Roughly speaking, according to the capabilities of the hardware, any pixel on the screen can be turned to any of these 216 colours at any time. This means that artists have much more freedom of colour selection.
On older hardware, the hardware's rendering ability was much more limited. Depending on the mode of operation, a limited subset of colours (known as a palette) could be used to render the screen. For example, the SNES had a 15-bit master colour palette, and based on that, you might think that the SNES is capable of outputting graphics of a similar colour depth to modern desktop computers, but not all of them could be used at once:
The Picture Processing Unit (PPU) used in the Super NES has a 15-bit RGB (32,768 color) palette, with up to 256 simultaneous colors at once.
However, while the hardware palette can only contain 256 entries, in most display modes the graphics are arranged into between 2 and 4 layers, and these layers can be combined using additive or subtractive color blending. Because these blended colors are calculated by the hardware itself, and do not have to be represented by any of the existing palette entries, the actual number of visible colors onscreen at any one time can be much higher.
The exact number depends on the number of layers, and the combination of colors used by these layers, as well as what blending mode and graphical effects are in use. In theory it can show the entire 32,768 colors, but in practice this is rarely the case for reasons such as memory use. Most games use 256-color mode, with 15-color palettes assigned to 8x8 pixel areas of the background.
If you want to know more, I recommend Retro Game Mechanics Explained video series on SNES graphics.