Why velocity potential? (Aero-)Acoustics (among other parts of fluid-dynamics) loves the velocity potential $\varphi$ defined as $\vec{v} = \nabla \ \varphi$ with the condition of $\nabla \times \ \vec{v} = 0$. I am definitely not trying to undermine this concept, I intuitively like that as well, but why exactly is that so cool?
I am aware of:


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*It's a scalar quantity with easy connections to acoustic pressure and velocity.

*It can be used as a descriptor of curl-free part of the field (using e.g. Helmholtz decomposition) when more complex fields are given.


If you would prefer more exact questions:


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*Given $\nabla \times \ \vec{v} = 0$, what is the equation that can't be formulated 'equivalent elegant' for acoustic pressure? (Which is scalar as well...)

*Are there any typical operations, derivations etc. that are significantly harder to be formulated for pressure than for velocity potential?

 A: 
(Aero-)Acoustics (among other parts of fluid-dynamics) loves the velocity potential 

It is something that has nothing to do with Acoustic or any other area where potentials come into play. Rather, it is a general property of fields provided certain assumptions hold.
For (usually) simply connected regions whenever you have an irrotational field, that is $\textrm{curl}\,\mathbf {v}=0$, then it exists a function $f$ such that $\mathbf{v} = \textrm{grad}\,f$ in every point $x$ where the domains of definition make sense. The function $f$ is then said to be a potential for the field $\mathbf{v}$. Likewise, knowledge of the potential function in any point of the space allows to derive back the field, equivalently (again, provided the correct assumptions on connectivity of the regions to hold).
This pretty much summarise the entire fairy tale about the potential in any area of physics where they come into play. It is always easier to solve equations involving scalar quantities rather than vectors, therefore whenever potentials may be defined, equations are usually rewritten in terms of the latter for the sake of simplicity. The original vector quantities can eventually be derived back taking derivatives thereof.
Notice on close that, nevertheless, there are some very special cases where, in physics, potentials and field are not completely equivalent (in the sense of invertible one from the other) in every point of space and time; this is mostly due to some very special topologies one deals with: the most remarkable example is the Aharonov-Bohm effect. But that is a corner case and other than that no special features appear when passing from vectors to their potentials except moving the complexity of the equations onto scalar ones.
