What makes the difference and what is the difference between open and closed bicycle wheels?

Speed, safety, and weight.

Speed: disc, aka closed wheels, are faster because they have less drag. You part the air better and with less disturbance.

Safety: why don't you see these on your average bike? I mean, after all, there are no spokes! Cross winds make the wheel a giant sail and can send you flying. Very dangerous.

Weight: They're heavier. I know I don't want to pedal a heavy wheel up a steep hill.

By closed vs. open I assume you mean disk vs. spoked? A disk wheel is much more aerodynamic than a standard spoked wheel. A high-quality disk is no heavier than a spoked wheel so there is usually not a big trade-off in weight. The catch is disks are very hard to control. They are illegal in mass start road races so can only be used in time trials and on the track. Even then you'll usually only see them on the rear. Some trackies will run them front and back.

What makes disks so difficult is they are extremely susceptible to wind. Even a slight breeze will make it very tough to maintain control. Plus, they do not corner well. They work in time trials because those are often straight (at the amateur level) and since you're riding alone the lack of control only affects you, not other riders. They are used in track racing when the velodrome is indoors. For outdoor tracks the same rules apply about the wind.

Their "arerodynamic profile" is far better than ordinary wheels. In other words they cut through the air more effectively than an ordinary spoked bike wheel because the air flows over the flat surface.

Enclosed wheels are more aerodynamic, but usually only used for indoor races, triathlons, or by certain enthusiasts, because they are very susceptible to cross winds that can throw you off balance or make turning difficult.

Closed assist the bike to go quicker.