If your objective is to simply obtain the equations of motion for the simple harmonic oscillator in 1d, there is little advantage in using a Lagrangian. However, in more complicated problems, this approach is much more powerful that Newton's law (see for instance this problem).
The Lagrangian is a scalar, which means it does not depend on the coordinate system (although of course its exact form will depend on the coordinate used). This makes the Lagrangian well adapted for problems where it would be otherwise difficult to add vectors. In particular, since kinetic and potential energies are additive, it is easy to construct the Lagrangian for a multi-particle system.
The equations of motion can be obtain from a single function: the Lagrangian. The Lagrangian is also efficient, in the sense that you do not need to consider action-reaction pairs; this makes it easy to handle constraints.
The resulting equations of motions are elegant: through calculus of variation, the solutions to the Euler-Lagrange equations minimize the action integral. The extension of the equations of motion from particles to fields is immediate.
Finally, the Lagrangian formulation is a stepping stone to the Hamiltonian formulation and its extensions (including Hamilton-Jacobi).
