CuriousOne's response is one of the most strikingly beautiful explanations I've seen. But to answer your other questions:
"Is the Path Integral formulation of QM just a mathematical tool or does it offer deep physical insights on the nature of QM? Is it just an alternate way to describe Quantum Mechanics?"
The word "just" is key here. If you don't agree that it's possible to adopt an intuition via some sort of explanation congruent to that of CuriousOne, then you are still able to observe the connection to classical dynamics offered by the path integral formulation. The Lagrangian formulation of mechanics via the principle of least action has been around for a long, long time and represents the classical limit (or $\hbar \rightarrow 0$ limit. see ref ) of the path integral. So, it is not just a mathematical tool- it is the extension of a concept that underpins classical mechanics- that, in itself, does indeed give insight into the deeper nature of the quantum theory.
"Could someone say that the schrodinger formulation is better/worse than the path integral formulation or are they just two different ways to describe the same thing?"
We can show that the reproduce the same equations of motions in the non-relativistic case. So in that sense they describe the same things. However, things don't go so well with the Schrödinger equation when we approach the relativistic case. The path integral approach does better in this regard.
"If the later is the correct answer, then why did the path integral formulation ever need to be developed? Does it offer something more that the original formulation does not offer?"
Besides offering a deep connection to classical physics, the path integral approach turns out to make it much easier for us to do calculations in quantum field theory. It is, however, more difficult to make rigorous formulations for this approach (you can google the mathematical difficulties of "constructive quantum field theories" to find all the info you need on that). Despite this, the answer is "yes": it offers a lot more than just describing the same phenomena in different ways.
Reference above: Paul A. M. Dirac, "The Lagrangian in Quantum Mechanics", Physikalische Zeitschrift der Sowjetunion, 3 (1933) 64–72}}