Okay...a game hen is a hen of a game type breed.
Traditionally, games were used for, well, gaming...cock fighting. So the roosters can be quite a handful, mostly to each other.
They are overall smaller, more refined birds, longer wings, lighter body. They tend to be flighty to very flighty, depending on the line, and because of their nature and body type can be excellent flyers (use netting and high fences).
Typically they are below average to poor layers of medium to small eggs.
They are usually kept for ornamental purposes or for brooding. Game hens often make excellent brooding hens (sitting on eggs to hatch chicks) and mothers.
Since they are very active and naturally cautious, there are those that keep them for open ranging as they do well in natural settings against predators.
A laying hen is more domesticated. They have been carefully selected for generations for laying, so the brooding instincts are often poor (depending on the breed and line). Since they are less wild, they are usually calmer (depending upon the line/breed).
They do not fly well. They have more of a "bumblebee" body style than the sleek "wasp" style of the game.
Layers are heavier than games, though not as heavy as the meat bird types.
Their talent is in egg laying, and the commercial lines do a stunning job of that.
Great question. You never know until you ask.
LofMc