RC filter transfer function and gain
The transfer function is not \$H(\omega)\$, it is \$H(j\omega)\$ (note the \$j\$, which makes it complex):

$$H(j\omega) = \frac{1}{1+j\omega RC}$$

This is important because the transfer function captures the phase in addition to the amplitude.

The amplitude is

$$|H(j\omega)| = \frac{1}{\sqrt{1 + (\omega RC)^2}}$$

by the definition of complex magnitude. This is the gain you measure from input to output and it may be all you care about, especially for a first order system. However, the phase $$\angle H(j\omega) = -\arctan(\omega RC)$$ can be quite important (e.g. for ensuring stability), especially for higher order transfer functions.

The main, direct use of the transfer function is to capture both gain and phase in one expression, but can also be used for time-domain analysis, as the transfer function is the Laplace transform of the impulse response. It is also useful for characterizing a multiple stage system, since the transfer function \$H(j\omega)\$ of a system consisting of stage \$H_1(j\omega)\$ followed by \$H_2(j\omega)\$ is simply \$H(j\omega) = H_1(j\omega)H_2(j\omega)\$ whereas in the time-domain you need to convolve the impulse responses \$h_1(t)\$ and \$h_2(t)\$ to find the overall system impulse response of \$h(t)\$.