What are the similarities/differences between the behaviors of Quantum particles and bouncing droplets? Bouncing droplets on a fluid surface show many weird behaviors of the quantum world. Look at this for example:
https://arxiv.org/abs/1307.6920
They can show tunneling, double-slit interference patterns, and even particle-antiparticle creation (when two solitary waves meet, according to the aforementioned reference). Some people take this as an approval of the pilot-wave interpretation.
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? And, is this a hint that quantum behaviors might have a classical underlying reason, similar to what happens to the droplets?
 A: 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 [12] 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 [15]. 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.
A: Don't believe the hype. Bouncing droplets tell us almost nothing about quantum mechanics.
The point of the bouncing droplet experiment is that, with some effort, you can set up a fluid system, one aspect of which is approximately described by the Schrodinger equation. 
This is not impressive, because there simply aren't that many simple linear partial differential equations: the heat, wave, Schrodinger, and Laplace equations basically exhaust the list. There are a million things out there besides heat which happen to obey the heat equation, but you don't see anyone talking about that because it's clearly coincidental -- not everything obeying the heat equation literally is heat. But the woo surrounding quantum mechanics means that you can get lots of press coverage if you make a system obeying the Schrodinger equation.
Furthermore, the really interesting, nontrivial features of quantum mechanics come in when you have multiple particles. You might get the impression, from news articles and Youtube videos, that the bouncing droplet system can simulate multiple quantum particles by just putting multiple droplets in. This is completely wrong. If quantum mechanics were really equivalent to just plain old classical particles interacting with a single classical field, then it wouldn't have any of the amazing, counterintuitive features we've seen it has, such as quantum entanglement. If this were really true, there would be no difference between classical and quantum computers, and no need for the scientific revolutions of the early 20th century that overthrew the fluid paradigms of the 19th.
In pilot wave theory, one can account for the quantum dynamics of $100$ particles by having them all interact with a classical field. But there's a catch. To reproduce all the complexities of quantum mechanics, that field has to live in an abstract $300$-dimensional space -- it is not anything like a familiar field in $3$-dimensional space. So while the bouncing droplet setup is a decent toy model for one particle, it's simply impossible for it to go any further.
