Understanding Segal's definition of conformal field theory

I have a fundamental problem in understanding Segal's definition of a conformal field theory:

On the one hand his monoidal CFT-functor is a formalization of the fact that, physically, the integrand of the time evolution path integral depends only on the conformal structure of the parameterization space $\mathbb{S}^1 \times [0,T]$ of the world sheet. More general, we want different time evolution operators on the state space for every bounded Riemann surface (after a Wick rotation)..

On the other hand, veeery roughly speaking, he looks at the group of conformal maps on the infinite cylinder, throws out one half and complexifies it. The result are again Riemann surfaces, but now with two parameterized boundaries. Now he says that a CFT is simply a representation functor of this categorified semigroup $A$, ensuring that the action of "one half of the group of conformal maps on the infinite cylinder" (namely Diff($\mathbb{S}^1$)) on the state space is of positive energy..

My question is now, how are these two ideas related? Also, why do I want a representation of the group of conformal maps at all? Conformal maps lead to equivalent conformal structures, which we don't care for (at least i thought so)...