Neo-classical images are traditional in lighting style and composition, but with a modern twist. These are still mostly forward facing posed portraits.
This is definitely the most "trendy" style of photography out there right now in my opinion. Most "natural-light" photographers shoot this style because of the relaxed nature and there is not as much (if any) equipment or backgrounds required.
Traditional portraits are just that: " traditional ". NOT boring, but classy. These are posed portraits with faces towards the camera. They are PROFESSIONALLY lit with classic PORTRAIT lighting.