In the picture below I described an RC low pass filter (H in Fourier, HL in Laplace domain). The input signal is a Sin[t]. I get that Fourier analysis should not get me the time transient response and indeed it's there only for t<0 (if it has any meaning is unclear to me) but why is the steady state response different from that of Laplace? Just look at the red boxes where I highlighted the t>>0 response. I tried with a Cos[t] and again there is a sign difference.