What is the difference between the Multiverse theory and the Many Worlds theory

There is no official multiverse theory.
There are just many hypotheses that say that there isn't one universe, but many (or infinitely many).

Many worlds interpretation is an interpretation of a quantum mechanical phenomena.
Quantum mechanics says that a state of a certain particle is unknown until measured.
It exists as a superposition of all of its possible states (wavefunction).
The act of measurement 'collapses' the wavefunction, and the state of the particle is known.
Many worlds interpretation states that the universe is constantly splitting into other universes in which the named particle is in its other condition.

Multiverse makes sense; many worlds does not.

Multiverse simply acknowledges the obvious; with an infinite and eternal mother universe that laid the low entropy egg we call the big bang, it is very unlikely that she has laid just one low entropy egg of space time. That is to say, even though laying a low entropy universe might be a rare event, the fact that the mother universe has both eternity and infinity to do that in makes even rare events expected.

Many worlds is not only not obvious, it's down right silly. It says that each event spawns multiple paths of follow on events. So as time is the unfolding of events, multiple time lines are being created continuously. And each time line, each sequence of events, is a different world. Using quantum physics with its reliance on probability distributions, math gurus can show how this is possible. But, see, this is a math model, not a physics model; and just because the math says it's possible, that doesn't make the physics.

If by "examples" you mean empirical evidence of either model, there are none. Both models are strictly WAG, hypotheses. Which means there are no falsifiable tests or observations to validate them.