What is the purpose of studying Sturm-Louville eigenvalue problem? After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential equations. 
Firstly, knowing a DE is Sturm-Louville system doesn't automatically produce a solution. Also, it does not give you the form of the eigenvalues. I would think these are the most important aspect of studying DE. 
What is done however, is to first solve the DE and then recognize it is a SL DE and hence satisfies a bunch of useful properties such as minimum eigenvalue, etc.
But even knowing all this, I have not been able to solve DE faster knowing that it is a SL DE or it satisfies a myriad of properties. Why do we characterize an ODE as SL problem at all?
 A: 
it does not give you the form of the eigenvalues. I would think these are the most important aspect of studying DE. 

Sure, it's great when we can cleverly manipulate an equation and produce explicit formulas for eigenvalues and eigenfunctions. Problem is, it almost never happens. (It happens in textbooks, for problems that are crafted to have explicit solutions. But not for problems that come from not-too-idealized models of the real world.) 
So, given that we can't solve differential equations explicitly, what should we do? We study existence, uniqueness, qualitative behavior (maximum principle, smoothness, etc), short- and long-term asymptotics. These help us better understand the thing we model with a differential equation, even without having an explicit solution. 
One source of Sturm-Liouville problems is the diffusion equation $u_t = (p u_x)_x $ and its relatives. Using the separation of variables, we find that there are solutions of the form
$$u(x,t) = \sum X_n(x)T_n(t) \tag1$$
where $X_n$ are eigenfunctions of a SL problem and $T_n$ are exponential functions with eigenvalues in the exponent. But is every solution of the form (1), or are we missing some? Turns out, all solutions are, because the eigenfunctions of SL problem form an orthogonal basis of $L^2$.  
Do solutions decay or grow as $t\to\infty$?  Depends on the sign of eigenvalues. 
Also, what is the approximate shape of  $u$ when $t$ is large? It should be the profile of the eigenfunction for the lowest eigenvalue. Ah, but is there one such eigenfunction, or do we have multiple ones? That is, do all solutions have similar profile in long term, or are there different scenarios? The study of multiplicity of eigenvalues ensues. 
Or maybe we want to figure out how quickly the higher eigenvalues leave the scene, being dominated by the lower ones. Enter the lower bounds for the spectral gap...  
