Heavier-than-air flight had been pursued for centuries before 1903. Especially in the decades immediately preceding 1903, there was something of a glider craze. Up until this point, the only viable means of flight was lighter-than-air, meaning balloons (whether hot-air or helium) or zeppelins (which were only patented a few years before).

As you saw in that thread, the military potential of the zeppelin was not lost on the Germans during the First World War. But balloons were appropriated for military purposes as early as the 1790s during the French Revolutionary Wars, and there was a specific section of the U.S. Army dedicated to balloons during the American Civil War. They represented a huge strategic advantage, allowing one to view enemy emplacements and troop movements from relative safety. Their only real disadvantage was their lack of mobility; they could not be quickly deployed or recalled. In the early months of World War I, they also presented nice targets for more accurate artillery and machine guns.

But before we get ahead of ourselves, the Wright brothers designed the Wright Model A, also known as The Military Flyer, specifically to demonstrate to the United States, Great Britain, France, and Germany. It was smaller and faster than previous Wright models, and several of these nations did purchase it, including the United States. This was the birth of modern, heavier-than-air military aviation.

The plane is far superior to balloons for the purpose of reconnaissance. It can fly higher, move faster, deploy and return more quickly (with the benefit of an airfield, of course). It isn't as susceptible to hostile ground fire, making it virtually impossible to keep above-ground operations secret to the enemy.

You're specifically asking about the differences between the Wright models and the biplanes and monoplanes used in World War I, which are very different by 1918. For my purposes, I will use the Model A Military Flyer as an example of a Wright plane. It utilized a push configuration, having the engine on the back and literally propelling the craft through the air. What we would think of as the tail was actually in the front. One of the first changes we see is the creation of a strong tail moved to the back; in many cases, this made the aircraft more stable by moving the center of gravity away from the nose. The next change is the tractor configuration, or the propeller at the front of the aircraft. Having the propeller at the front makes the aircraft more maneuverable mainly because the pusher configuration created extra drag. The final death knell for the pusher configuration was the development of the interrupter gear that allowed machine guns to fire through the tractor configuration without damaging the propeller. The structures of these aircraft remained wood and canvas until late in the war. This was because the materials were light and the engines of the time were unable to generate enough power to lift anything too heavy. But as the engines improved, the use of metal for the fuselage increased.

Sources:
Hampton, Dan. Lords of the Sky: Fighter Pilots and Air Combat, from the Red Baron to the F-16. New York: HarperCollins, 2014.
Hallion, Richard P. Rise of the Fighter Aircraft: 1914-1918. Annapolis: 1984.
McCullough, David. The Wright Brothers. Simon & Schuster, 2015.

The early planes in WWI were not dissimilar to the Wright Flyer. Some of them were pusher prop planes with small fuselages*. In the interim (since aircraft were being researched in other countries as well) advances by Bleriot (monoplanes), for instance caused developments. At the very beginning of World War I, aircraft were mainly observer platforms and more or less crude compared to late war planes and four years of accelerated wartime development.

• sort of a bigger deal, having framework and canvas or wood hulls.