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Let 
$\phi $ and 
$\psi $ be analytic maps on the open unit disk 
$D$ such that 
$\phi (D) \subset D$. Such maps induce a weighted composition operator 
$C_{\phi ,\psi }$ acting on weighted Banach spaces of type 
$H^{\infty }$or on weighted Bergman spaces, respectively. We study when such operators are order bounded.
We consider differences of composition operators between given weighted Banach spaces 
or Hv0 of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results.
Every weakly compact composition operator between weighted Banach spaces 
$H_{v}^{\infty }$ of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space $H_{v}^{\infty }$ are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.
