What category are you working in? Operads are definable in any symmetric monoidal category.
They always have associated monads in that category such that the categories of algebras over
the operad and monad are isomorphic. That relationship between operads and monads motivated
my coining of the word "operad'', back in 1971. The original definition was for spaces, but
Max Kelly and I understood the general situation right away. It is rare and special that the
monad arising from an operad is cartesian, so it is not true that ``from every operad arises a
cartesian monad''. Incidentally, it is not only true that different operads can have the same
associated monad, it is also true that the same operad can have different associated monads in
different categories. For example, an operad $\mathcal{C}$ in spaces with $\mathcal C(0)$ a point has
a monad in unbased spaces and a quite different monad in based spaces with isomorphic categories
of algebras.

With my original definition of an operad, not every cartesian monad arises from an operad.
For a perhaps esoteric example, consider the monad in the category of globular sets that
defines strict $\omega$-categories.