I`ll try a short answer.
* Solving the 2nd order diff. equation in the time domain you calculate the transient behaviour resulting in an expression exp(sigma*t)*exp(jw*t)=exp(sigma*t+jw*t). The 2nd term gives the frequency and the 1st term gives the amplitude (decreasing for sigma<0). Thus you get a transient behaviour equal to a decaying sinusoidal wave. This approach leads to the definition of a complex frequency variable s=sigma+jw and exp(s*t).
* All parameters and properties of a system in the frequency domain are defined for steady-state conditions only. That means: constant amplitudes of the sinusoidal signal (assuming that all transients have disappeared).
*That means: sigma=0 and s=jw (remember: steady-state conditions with rising/falling amplitudes are not possible; it is a contradiction).
Hope this answers your question.
---------- Post added at 12:37 ---------- Previous post was at 12:29 ----------
Additional remark: On the other hand, it is common practice to use the complex variable s=sigma+jw also in the frequency domain - however, only for theoretical purposes (e.g. pole/zero locations, definition of pole parameters like Q and pole frequency). Thus, it is very easy and illustrative to describe, for example, filter properties. However, if you speak about the real frequency response, which can be measured or simulated, you always set sigma=0.