If heat (or thermal energy) are vibrations of particles and sound is a wave that is propagated through medium e.g vibration of air particles, what indicates if vibration of particles will be perceived as sound or heat (what is the difference between these vibrations) ?
Heat corresponds to random movements of atoms and molecules. It travels only through conduction - slowly. Sound consists of ordered movements, travelling through a medium as a wave (although it can also stand still, as in a standing wave). Large numbers of atoms or molecules move back and forth in synchrony. Sound eventually becomes random, as it is scattered around in many directions, and ultimately ends as heat.
A qualitative picture of what happens in a gas can be made in terms of whether the behavior is random or non-random, oscillatory or steady.
Temperature describes the random motions of the particles that comprise some object. Correlations, if they exist, disappear rapidly with distance between particles. In an ideal gas, correlations don't exist, period.
Sound and gas flow (e.g., wind) are non-random behaviors. Behaviors between particles are correlated, and these correlations don't dissipate nearly as quickly as they do with temperature. The difference between sound and wind is that the behavior is oscillatory in the case of sound but steady in the case of a wind.
If heat (or thermal energy) are vibrations of particles and sound is a wave that is propagated through medium e.g vibration of air particles ...
This one sentence displays a number of misunderstandings.
Writing "heat (or thermal energy)" represents one of those misunderstandings. Do not confuse heat and thermal energy! Objects do have thermal energy, but they do not have "heat".
Consider what happens when an object as it is taken from some initial thermodynamical state to some final thermodynamical state. The change in the object's thermodynamical energy depends only on the initial and final states. How the object is taken from the initial state to the final state is irrelevant when it comes to energy. That is not the case with regard to heat and work. Here the path from the initial state to the final state dictates whether the energy transfer is in the form of work, heat, or both. It's erroneous to say that objects contain "heat" (or "work", for that matter) because work and heat are path dependent quantities.
The next problem is "vibration of particles." While this is how you need to look at a solid to understand temperature, this does not give a good picture of a gas. The concept of an ideal gas is the right place to look to gain an understanding of the thermodynamic behavior of a gas. Near-ideal gases have very small particles. Collisions occur between particles and the walls of the container of the gas; all such collisions are elastic. Ideal gases have extremely small particles; ideally they are point masses. Point masses don't collide with one another. They only collide with the walls of the container.
This simple picture of an ideal gas yield a very simple description of temperature and pressure as a consequence of the average speed of the point masses that comprise the gas. While that assumption of point masses is non-realistic, it can be amazingly accurate at times. Kinetic theory can be extended to some extent to describe real gases, where the particles aren't point masses. The place to start your studies is with an ideal gas, and then relax some of those overly simplistic assumptions. Most introductory calculus-based physics and introductory calculus-based chemistry texts do just that.
There's not much difference. Thermal vibrations would be perceived as sound (noise) if they were intense enough, but they are not. The thermal vibration amplitudes at room temperature are small enough that the ear is not sensitive to them.
I've been told that the sound pressure level for thermal vibrations is close to, but below, the threshold of hearing, and that evolution proceeded in such a way as to suppress what would be noise everywhere. I can't verify that ... could be a myth.