In systems theory and signal processing, we often transform expressions based in the Laplace $s$-domain into the complex frequency domain with $j\omega$ (engineering notation for the angular frequency on the imaginary axis): $s \leftrightarrow j\omega$.
I have always been told I can easily transform expressions between the two, but that I shouldn't simply equate them. So far, I always simply considered the Laplace $s$ as a 'sort of frequency or pulsation'. Is this correct?
What exactly is the difference in meaning between these two domains?
PS: Feel free to (re)tag, as I am not familiar with this particular forum's tags.