Many sources state that the Laplace transform is simply the Fourier transform where $\text{Re}(s) =0 \implies s = j\omega$
However, why is it that substituting $s = j\omega$ into Laplace transforms of a function $f(t)$ do not yield the corresponding Fourier transform (for large number of transforms)?
For example,
The laplace transform of $\sin(\omega_o t)$ is $$\dfrac{\omega_o}{s^2 + \omega_o^2}$$
(According to source)
Where as the fourier transform yields:
$$i\pi (\delta(\omega+\omega_o) - \delta(\omega-\omega_o))$$
(According to source)
But evaluating $$\dfrac{\omega_o}{s^2 + \omega_o^2}|_{s = i\omega} = \dfrac{\omega_o}{-\omega^2 + \omega_o^2}$$
Quite the stark contrast.
When does the Fourier transform and the Laplace transform coincide for $s = j\omega$
Why is that sometimes they do not coincide?