Very roughly - In the many worlds quantum physics theory a measurement in the quantum realm are interpreted as a particular World. Therefore the wave function of quantum physics is not interpreted as probabilities but as many worlds. These worlds are sometimes called parallel universes.
The Multiverse, another instance of using the term many worlds, are other universes beyound our universe. These many worlds are not necessarily a quantum feature but a cosmological feature.
In Philosophy like many worlds you have the term Possibly Worlds of Modal Logic. Modal logic involves study of Necessary and Possible in logic.
So the term "Many Worlds" can mean very different things. When Galileo first saw mountains on the moon, some natural philosophers postulated a Plurality of Worlds where other intelligents lived. Leibniz talked of many worlds or possible worlds that God could create to help explain his philosophy. Later inspired by Leibniz, philosophers while working with Modal logic began using possible worlds.
There are several meanings to "Many Worlds" so you must differentiate. The quantum many worlds are in the same time and space so can effect each other. The Possible worlds of modal logic are not in the same time and space so can not effect each other.
The Possible worlds of modal logic are seriously studied by philosophers.
The quantum many worlds are seen by many physicists are fringe science.
The multiverse in speculative. The multiverse unlike possible worlds in modal logic is connected by time and space.
The Plurality of Worlds from Galileo's time is widely held to be true by todays scientists. Astronomers are discovering many new extra solar planets and believe that in our galaxy there must be worlds which are inhabited by life.
The "Possible Worlds" theory of Leibniz is taken seriously by philosphers who study big questions like the existence of evil. Logicians are also interested in Leibniz "Possible Worlds".
Mixing up the types of "Many World" studies especially between philosophy and physics can get confusing and lead you down the wrong path. A thinker should admit what he doesn't know while still being able to differentiate some high level concepts like "Many Worlds".