In the naval sphere battleships were the weapon of choice in World War One, but they were made obsolete by the introduction of aircraft carriers, which were an essential part of the American war effort in the Pacific. Sonar was another invention that helped the Allies defeat Axis submarine warfare.

Along with new and better aircraft, World War Two saw the first use of guided missiles like the V-2 rocket, which was a precursor to the ICBMs and cruise missiles of today.

Another important weapon not yet in existence in World War One was bacteriological warfare. During the Cold War both the U.S. and the Soviet Union amassed stocks of anthrax and other microorganisms that were to be used in the event of a global conflict. Unlike most other new weapons, however, they have not yet been openly used.

WWI is unique in that industrial processes were used to make new weapons.  Poison gases that were otherwise byproducts of industry were used on the Western Front as were belt-fed machine guns, tanks, and barbed wire.  

For the sake of your question, I will compare WWI with WWII to give you an idea of how weaponry changed.  First, there was no such thing as the personal machine gun in WWI--machine guns often had a team of at least three people--someone to aim and fire, someone to keep the barrel cool with a bucket of water, and someone to ensure that the belt of ammunition fed in smoothly.  The average WWI infantryman carried a bolt-action Mauser, Enfield, or Springfield rifle, depending on which side he represented.  In WWII, men could carry their own automatic weapons--the Soviet Union carried weapons that were similar to the AK-47.  WWI flyers also had very primitive planes that did not have aluminum frames--in WWII, aluminum would be a key airplane component.  WWII naval craft also used sonar--in WWI, this technology did not exist yet.  Of course, atomic weapons and jets did not exist in 1918 either.