What's the point of Hamiltonian mechanics?
There are several reasons for using the Hamiltonian formalism:
Statistical physics. The standard thermal states weight of pure states is given according to
$$\text{Prob}(\text{state}) \propto e^{-H(\text{state})/k_BT}$$
So you need to understand Hamiltonians to do stat mech in real generality.
Geometrical prettiness. Hamilton's equations say that flowing in time is equivalent to flowing along a vector field on phase space. This gives a nice geometrical picture of how time evolution works in such systems. People use this framework a lot in dynamical systems, where they study questions like 'is the time evolution chaotic?'.
The generalization to quantum physics. The basic formalism of quantum mechanics (states and observables) is an obvious generalization of the Hamiltonian formalism. It's less obvious how it's connected to the Lagrangian formalism, and way less obvious how it's connected to the Newtonian formalism.
[Edit in response to a comment:]
This might be too brief, but the basic story goes as follows:
In Hamiltonian mechanics, observables are elements of a commutative algebra which carries a Poisson bracket $\{\cdot,\cdot\}$. The algebra of observables has a distinguished element, the Hamiltonian, which defines the time evolution via $d\mathcal{O}/dt = \{\mathcal{O},H\}$. Thermal states are simply linear functions on this algebra. (The observables are realized as functions on the phase space, and the bracket comes from the symplectic structure there. But the algebra of observables is what matters: You can recover the phase space from the algebra of functions.)
On the other hand, in quantum physics, we have an algebra of observables which is not commutative. But it still has a bracket $\{\cdot,\cdot\} = -\frac{i}{\hbar}[\cdot,\cdot]$ (the commutator), and it still gets its time evolution from a distinguished element $H$, via $d\mathcal{O}/dt = \{\mathcal{O},H\}$. Likewise, thermal states are still linear functionals on the algebra.
Some more comments to add to user1504's response:
For a system with configuration space of dimension $n$, Hamilton's equations are a set of $2n$, coupled, first-order ODEs while the Euler-Lagrange equations are a set of $n$, second-order ODEs. In a given problem it might be easier to solve the first order Hamilton's equations (although sadly, I can't think of a good example at the moment).
It's true that quantum mechanics is usually presented in the Hamiltonian formalism, but as is implicit in user1504's answer, it is possible to use a Lagrangian to quantize classical systems. The Hamiltonian approach is commonly referred to as "canonical quantization", while the Lagrangian approach is referred to as "path integral quantization".
Edit. As user Qmechanic points out, my point 2 is not strictly correct; path integral quantization can also be performed with the Hamiltonian. See, for example, this physics.SE post:
In Path Integrals, lagrangian or hamiltonian are fundamental?
First of all, Lagrangian is a mathematical quantity which has no physical meaning but Hamiltonian is physical (for example, it is total energy of the system, in some case) and all quantities in Hamiltonian mechanics has physical meanings which makes easier to have physical intuition.
In Hamiltonian mechanics you have canonical transformations which allows you change coordinates and find an easier canonical coordinates and momenta in which it is easier to solve problem.
The best of all is, Lagrangian is a powerful mathematical method to solve problems in classical mechanic but Hamiltonian is a powerful method to solve problems in classical mechanics, quantum mechanics, statistical mechanics, thermodynamics... etc actually almost all physics...
For example: In thermodynamics: Gibbs free energy, Helmholtz free energy... are all canonical transformations of Hamiltonian..