Is the Laplace transform a special case of Fourier transform? (Not the other way around)
The Fourier Transform is the Laplace Transform with the complex variable s restricted to be the imaginary axis on the s plane. For this reason the Fourier Transform only exists when the imaginary axis is within the region of convergence. The variable s is called a "complex frequency" as it is the frequency variable that can take on real ($\sigma$) and imaginary ($\omega$) components. That said, I would view the Fourier Transform as a subset of the Laplace Transform, or the Laplace Transform as an expansion on the Fourier Transform that provides a lot more functionality and can exist when the Fourier Transform can't. 

This is also the reason that the frequency response for a system with a general transfer function $H(s)$ is given as $H(j\omega)$. 

When a system is restricted to $s= j\omega$ as the input, then the input is restricted to be only sinusoids or signals given by $e^{st}$ with $s = j\omega$ which maintain a constant magnitude with time. By allowing s to have real and imaginary components as in $s = \sigma + j\omega$ then we also allow the input to grow or shrink with time, depending on which point in the s plane is used as the input to the system.