Given an adjunction, we get a monad on one side and a comonad on the other side. What is the connection between their algebra and coalgebra categories? Are they allways equivalent?

The example i have in mind is the starting point of algebraic geometry (or more general: The fundamental theorem of formal concept analysis) ~ the relation between polynomials and points in affine space given by fRx iff f(x)=0 induces an order preserving isomorphism between radical ideals and "algebraic sets".