The asteroid belt and the Kuiper belt are based on observations, the Oort cloud is theoretical. The Kuiper belt and Oort cloud are both believed to be made of frozen material.
The asteroids belt is an observed section of the universe between Mars and the gas giants. The asteroids are made of solid materials. The asteroids are thought to be rocks.
The Kuiper belt is made of observed proto planets. The most famous of these is Pluto. For years Pluto was thought to be a planet. Based on the observations of other small frozen balls of matter in the same elliptical orbits Pluto was demoted from its planet status.
The Oort cloud is thought to be made of the same type of frozen material as the Kuiper belt. The existence of the cloud was theorized because of the appearance of new comets, and the observed death of old comets. If the comets were created at the same time and processes as the solar system then the comets had to exist somewhere outside of the observable solar system.
The comets we can observe have a life span of less than 10,000
years. There is presently no empirical evidence for the Oort cloud but if the solar system is 4.5 billion years old something along the lines of the Oort cloud must exist.
The similarities between the Oort cloud and the Kuiper belt are the material they are thought to be made of. The Astroids and Kuiper belt are based on empirical observations while the Oort cloud is not.