Both the Laplace and the phasor (Euler) approaches are frequency domain solutions to time domain problems (essentially linear differential equations) The Laplace works in a complex frequency domain where s= sigma
+jw and leads to a complete solution to problems such as the one you propose, giving the transient response, as well as steady state DC and AC responses (including multiple frequencies. The phasor model considers the steady state response at one particular frequency corresponding to a single point on the sigma=0 axis.
Consider a series RLC circuit with 0 initial conditions and a voltage applied at t=0 v(t)=Ri(t) +Ldi(t)/dt +(1/C) integral of i(t) dt and knowing v(t) we can solve for i(t) If we use Laplace this results in V(s)=(R+sL +1/sC)I(s) or I(s)= V(s)/(R+sL+1/SC) and knowing v(t) we can find V(s) and solve for I(s) THEN convert back to the time domain through the inverse transform. We will get a complete solution for all times >t=0.
IF the applied voltage v(t) is a sinusoid, and we are only interested in the steady state response we can simply substitute jw for s. V(jw)=(R+jwL +1/jwC)I(jw)
There is a bit more math background here but we don't need to go into it at present.
The point of both is that it is easier to solve algebraic equations using complex numbers than it is to solve differential equations- particularly when you may have simultaneous equations. Neither phasor not Laplace voltages and currents exist except mathematically but they are convenient. The phasor approach is easier than the Laplace because you are looking for only a specific part of the total solution.
In your particular case, you are applying a sinusoid to a circuit which has a transfer function Vo/Vi =1/(1+sT) for any actual signal (within limits). In particular, for a sinusoid of any particular frequency, Vo/Vi=1/(1+jwT)in steady state.