I only know enough about fluid dynamics for a short observation:
From the cited paper:
Quantum theory in a hydrodynamic form was formulated by Erwin Madelung in 1926  as an alternative formulation of the Schroedinger equation. Remarkably that Madelung's equations exhibit a close relationship through the Bohmian mechanics [13, 14] with hydrodynamic equations such as the Navier-Stokes equations. It gives the reason to hypothesize that quantum medium behaves like a fluid with irregular fluctuations . On the other hand, we may suppose that behavior of an incompressible liquid can be described.
Understandably, a large amount of the paper cited is given up to the problem of the conversion of a notoriously intractable non linear differential equation, into the (lovely and simple :) linear Schroedinger equation.
What is the reason for this similarity of behavior between bouncing droplets and quantum particles? Is there any difference, where this analogy no longer works and droplets CANNOT mimic quantum particles?
When you read the paper, its impossible not to see that a lot of assumptions and suppositions are (understandably) made in order to get droplets to act, in a math like sense, as elementary particles.
I only say this because, although it's a very neat and ingenious paper which I would be happy to be capable of writing myself, I wouldn't read much more into it than that: i.e. a (fairly forced, imo) analogy between two different systems.
And, is this a hint that quantum behaviors might have a classical underlying reason, similar to what happens to the droplets?
Better people than I can give you a more comprehensive answer, but quantum behavior, as I'm sure you are aware, is more accurately described by Quantum Field Theory and I think the link between that theory and the classical world is wider in many ways than between the classical world and Quantum theory, particularly (obviously, sorry) in the relativistic realm.
Is there any difference, where this analogy no longer works and droplets CANNOT mimic quantum particles?
Yes, in two particular cases:
I think you would have severe difficulties reconciling quantum entanglement and also the interactions between particles using for example, the QED model, with the classical droplet's described above.
I hope you get better answers than this, but personally I would take it merely as an interesting coincidence that this particular classical system can mimic a quantum one.