I'm reading on the calculus of variations, and Hamiltonian mechanics is discussed there. However, I'm not a physicist, so I cannot very well place the concept of Hamiltonian Mechanics in perspective.

Wikipedia treats Hamiltonian Mechanics in a very formal way.

That's why I hope someone can explain to me what is the essential insight, or the essential change in approach, of Hamiltonian Mechanics.

I'd like to understand conceptually-historically what the change in mathematical approach is compared to Newtonian mechanics, what the change in intuition is, the insight that precipitates it, and why it is significant.

Edit: This question is different from this one, in that I'm asking more for the conceptual (and historical) significance and change of perspective of Hamiltonian Mechanics, rather than what I need to do to learn it.