To answer the second part of your question, I will give a classic example of harmonic motion. The potential energy of a spring is $U=\frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement.
Newtonian Mechanics:
$F=m\frac{d^2x}{dt^2}=\frac{-dU}{dx}=-kx$
So $m\frac{d^2x}{dt^2}=-kx$, which is an easy differential equation.
Lagrangian Mechanics:
First we know the Euler-Lagrange Equations $\frac{\partial L}{\partial q} = \frac{d}{dt}\frac{\partial L}{\partial \dot{q}}$, we identify coordinates $q=x$, and we define our Lagrangian $L=T-V$ ($T$ is kinetic energy and $V$ is potential energy).
$T=\frac{1}{2}m\dot{x}^2$
$V=\frac{1}{2}kx^2$
So we plug this all into our little Euler Lagrange Equation, and, solving through, you get (drum roll), $m\frac{d^2x}{dt^2}=-kx$!
Conclusion
So after all of this we get the same equation as with Newtonian mechanics and with a lot more work right? In this example probably, and most other simple systems. However, Lagrangian Mechanics has some very powerful applications.
Consider the following system: You have multiple pendula connected by springs, and each pendulum begins with some initial position and velocity. How do you go about solving this system? In Newtonian Mechanics it will get extremely complex to work out all the forces involved. 37
However, taking a Lagrangian perspective, much of the hard work is taken care of as you can easily define $q_i=\theta_i$, $\theta$ being the angular displacement of each the pendulum. And instead of having to deal with the various forces you just deal with the potential and kinetic energy.
An even more, exponentially more, important application is in classical field theory (I know it has some important connections to with QFT, but I am not in any position to knowledgeably comment on that). Electromagnetism and General Relativity are two excellent examples. You can derive <PERSON>'s equations entirely from the electromagnetic Lagrangian ( $L=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+A_{\mu}J^\mu$) and you can derive extremely important results in general relativity from the Hilbert Action ($S_H=\int \sqrt{-g}Rd^nx$) and similar variational principles. 800