I guess this question might spawn a very large spectrum of ansewrs, i'll try to take on the (simple) mathematical side of dealing with a system of first order ODE instead of an higher order one.
In this sense (as in many others!) the hamiltonian formulation of a dynamical problem, like the one you have posed as an example, gives in an easier way plenty of information that the lagrangian or newtonian (as $F=ma$) do not.

The reason i'll give you at first sight will look as purely mathematical but as it turns out it gives clear physical insights that the other forumluation do not give. I will denote as the n-th derivative of the function $x(t)$ the symbol $x^{(n)}$, Given an ODE written in it's normal form $$\tag 1 x^{(n)}=\mathcal{F}\left(t,x'...x^{(n-1)}\right) \\ \mathcal F:A\subseteq \mathbb R\times \mathbb R^n\longrightarrow \mathbb R^n$$ if we define the variables $x_{k}=x^{(k)}$ so $x=x_1,\ x'=x_2 ,\ x''=x_3,\ ecc \ $ than we can always write it in the form: $$\tag 2 x=x_1 \\ x_1'=x_2\\x_2'=x_3\\ . \\ . \\ .\\x_n'=\mathcal F(t, x_1,x_2,...., x_n) $$ Now, since dynamical problems are of the form $$\tag 3 m \ x''(t)=\mathcal F(t,x,x')$$ the problem can be broken in a system of two first order differential equations: $$\tag 4 \begin{cases}x'_1=x_2 \\ m \ x'_2=\mathcal F(t, x_1, x_2) \end{cases}$$

Ok, let's define the differentiable curve ${x}(t)\in C'(I\subset \mathbb R,\mathbb R^n)$ and the matrix $\mathrm A \in GL(\mathbb R^n)$ such that we can write the cauchy problem $$ \tag 5 \begin{cases} x'(t)=\mathrm A \cdot x(t)\\ x(t_0)=\xi_0 \end{cases}$$ long story short you can write the soultion as $$\tag 6 x(t)=e^{\mathrm A(t-t_0)}\cdot\xi_0$$
(There is a simple theory on how you can diagonalize and write the matrices $\mathrm A$ and $e^{\mathrm A t}$ depending on the nature of it's eigenvalues) Once you know the eigenvalues of $\mathrm A$  you can determine the asymptotic behaviour of the system: like it's stability ,if the soultion is limited and in some cases even the presence of  actractors just by knowing the eigenvalues of $\mathrm A$. ( the Hamiltonian matrix you mention)

The Hamiltonian forumlation of mechiancs is the most natural way to write Newton's law, a second order differential equation, into a system of two first order equations. Thus for the linearized theory and even for the case $x'(t)=\mathrm A\cdot x(t)+g(t,x)$  where $g(t,x)$ is "small" respect to $x(t)$ (and under different assumptions, for more general and more complicated cases too!). You can write the soultion as $$\tag 7 x(t)=e^{\mathrm A(t-t_0)}\cdot\xi_0+\int_{t_0}^t e^{\mathrm A(t-s)}\cdot g(s,x(s)) \ ds$$ and extract alot of information from the matrix $\mathrm A$ eigenvalues, as i already enphasized. 

The morale of the story is that, in mechanics the natural framework to carry on these very fast analsys (and more sophisticated ones too!) on the nature of the soultions  is the hamiltonian forumlation.

For example:

 When you write the matrix for $$\tag 8 \Phi''=-\omega^2 \Phi$$ and get $$\tag 9 
\frac{d}{dt} \left[ \begin{array}{c} X \\ Y \end{array} \right] =
\omega^2 \left[ \begin{array}{cc} 0 & 1 \\ -1 & 0\end{array} \right]
\left[ \begin{array}{c} X \\ Y \end{array} \right].
$$
it's easy to compute the eigenvalues which are pure immaginary $\lambda_{\pm}=\pm i \omega$ the theory tells you right away that the soultions are limited for $t \rightarrow \infty$ and now (knowing a cuple of theorems) you can write just by knowing the eigenvalues: $$\tag {10} \mathcal U(t) \equiv e^{\mathrm A t}= \left[ \begin{array}{cc} cos(\omega t) & sin(\omega t) \\ -sin(\omega t) & cos(\omega t)\end{array} \right] \\ S(t)\equiv \left[ \begin{array}{c} X \\ Y \end{array} \right](t)= \mathcal U(t) \left[ \begin{array}{c} X \\ Y \end{array} \right](0) $$ so $$\tag{11}||S(t)||=\sqrt{X^2(0)+Y^2(0)}=constant$$ i.e. the soultions are circular orbits! In less trivial exemples those kinds of analysis really saves you alot of time in my personal experience.