What is the significance of s-domain?
The \$s\$-domain or frequency domain is not really what is special.  It is the Laplace transform that is special.  With appropriate assumptions, Laplace transform gives an equivalence between functions in the time domain and those in the frequency domain. 

Laplace transform is useful because it interchanges the operations of differentiation and multiplication by the local coordinate \$s\$, up to sign.  This allows one to solve ordinary differential equations by taking Laplace transform, getting a polynomial equations in the \$s\$-domain, solving that polynomial equation, and then transforming it back to the time domain.  Likewise, ordinary differential equations in the \$s\$-domain correspond to polynomial equations in the time domain.

I'll just work out the canonical easy example here for illustration.

Solve the ordinary diff eq: 
$$f'(t) = f(t), f(0)=1$$

Let \$L\$ denote the Laplace transform. The theorem about Laplace transform we will need to use to switch differentiation and multiplication is
$$L(f') = sL(f)-f(0).$$ To begin solving our differential equation, we take the Laplace transform of both sides
$$L(f') = L(f)$$
which turns it into the polynomial equation
$$sL(f)-f(0) = L(f).$$

Solving for \$L(f)\$ we find that
$$L(f) = \frac{1}{s-1}.$$

The inverse Laplace transform gives us that
\$f(t) = e^{t}\$.

We check that this does satisfy our original diff eq
$$(e^{t})' = e^{t}, e^0=1$$