It is very important to distinguish between vibration and oscillation.
Vibration is basically defined as mechanical oscillation.
Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point.
Now you are asking what the difference between temperature and sound is.
You are assuming that both temperature and sound are caused by vibrational motions of the molecules (let's assume air in your case.)
A molecular vibration occurs when atoms in a molecule are in periodic motion while the molecule as a whole has constant translational and rotational motion. The frequency of the periodic motion is known as a vibration frequency, and the typical frequencies of molecular vibrations range from less than 10^13 to approximately 10^14 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm^−1.
Now the misconception is, that in air, you assume, that vibrational motions of the molecules is dominant. In reality in gases, translational and rotational energies of the molecules is dominant, and this is what we identify with internal energy (temperature) in the gases. Vibrational motion of the gas molecules is minimal relatively.
Finally, you were right: in most solids, vibration and translation are prominent in internal energy, while in common gases, translation and rotation are prominent while vibration is negligible. Yet, in both cases, relatively complex models are needed to explain what is measured experimentaly.
vibrational motion in gases
In solids, it is the opposite. Vibrational motions of the molecules are dominant in determining temparature.
But you are talking about sound, and you are assuming air.
Now you are asking why we hear sounds, but not the translational and rotational motion of the air molecules (not vibrational).
In physics, sound is a vibration that typically propagates as an audible wave of pressure, through a transmission medium such as a gas, liquid or solid.
You are correct to say, that sound is defined as a vibration that propagates as a wave in air.
Humans can only hear sound waves as distinct pitches when the frequency lies between about 20 Hz and 20 kHz.
In air, sound only propagates as a longitudinal wave, and it is important to understand that the particles of air themselves do not propagate with the sound.
Now when we hear with our ears, we are able to hear waves of air, that is our ears can hear from 20 Hz to 20 kHz. Internal energy (temperature) vibrations are usually between 10^13 to 10^14 Hz.