What is difference between transfer function and gain?

Answer 1

Gain is also taken as Laplace transform of output to Laplace transform of Input . for example voltage gain calculation , it is not necessary to make the energy will be zero in L and C ( if present in the given circuit). But in case of Transfer function to avoid the system dynamics , we have to make the inductor and capacitor energy will be zero as initial condition = 0

Answer 2

The gain is part of the transfer function and is the change in output over the change in input. A transfer function basically simplifies a system and its form depends on whether the response has a time delay, whether it's first or second order, etc