In the older picture, The polygonal shapes that form Mario's body are incredibly obvious. His gloves are nearly diamond shaped. His eyes, eyebrows, buttons and mustache are flat textures, and the rest of his body is textured with solid colors.
In the newer picture, his eyes, eyebrows, buttons and mustache are modeled, rather than being seemingly painted on. His gloves are detailed and his fingers are individually modeled and animated. While some polygonal edges are occasionally noticeable, generally, his form is rounded and smooth. The texture on his overalls goes from a flat shade of blue to detailed denim and the buttons have some interesting shaders applied.
One thing I've noticed in many newer games is that as characters get more realistic, colors tend to get more muted. This is also evident in these two pictures of Mario and is largely a stylistic choice.