  Algebras simple with respect to an action of a Taft algebra $H_{m^2}(\zeta)$
deliver an interesting example of $H$-module algebras that are $H$-simple but
not necessarily semisimple. We describe finite dimensional
$H_{m^2}(\zeta)$-simple algebras and prove the analog of Amitsur's conjecture
for codimensions of their polynomial $H_{m^2}(\zeta)$-identities. In
particular, we show that the Hopf PI-exponent of an $H_{m^2}(\zeta)$-simple
algebra $A$ over an algebraically closed field of characteristic $0$ equals
$\dim A$. The groups of automorphisms preserving the structure of an
$H_{m^2}(\zeta)$-module algebra are studied as well.
