Under the precondition of linearity you are allowed to treat a sum of different sinusoidal inputs separately.
Example: Vin1: cos(wt) and Vin2=j*sin(wt).
Thus, according to Euler: Vin=Vin1+Vin2=exp(jwt).
The ouput of a linear system also is a function of exp(jwt) and - in addition - it contains a phase shift exp(j*phi).
You always can come back to Vin1 or Vin2 by using the imaginary or real part of Vin.
The advantage of this (seemingly complicated) approach is that mathematical treatment of exp(jwt) is much more simple than using sin- oder cos-function (Fourier, Lapalace, Taylor-expansion,...).
More than that, the introduction of exp(jwt) opens the way to define the complex frequency variable s=sigma+jwt as well as the complex transfer functuion H(s). The usage of these complex variables and functions has many, many advantages for analog signal processing.
Therefore, both expressions - (H(jw)) and H(exp(jwt)) - belong to the same frequency response - it is only another way to express the same thing because the frequency response H(jw) can be derived from the transfer function H(s) simply by replacing s by jw.