A full course on Sturm-Liouville is worthy of being given in my humble opinion. As already mentioned, the Sturm-Liouville problem is connected with many problems from pure and applied mathematics. For examples,

- PDE, boundary problems, notion of Green function (as mentioned above)
- some special functions and orthogonal polynomials
- problems with ordinary differential equations (see the relevant chapters in Coddington-Levinson or in Whittaker): existence, uniqueness, explosion
- problems with eigenvalues, their asymptotics, introduction to the theory of self-adjoint operators, WKB approximations, ...
- the Weyl-Titchmarsh theory and its relation with complex analysis and Green's functions
- numerical computations of the eigenvalue problems or of the solution (shooting methods, ...), as a way to introduce numerical analysis.
- parabolic equations and links between semi-groups and resolvent (see e.g. ``The abstract Cauchy problem and Cauchy's problem for parabolic differential equations'' by E. Hille)
- Feller theory of one-dimensional stochastic processes is also strongly related to SL problems
- the Krein theory of the string (already mentioned), harmonic analysis, ...
- and of course the many applications of the SL equation in physics and other... (on the history, see ``Sturm and Liouville's work on ordinary linear differential equations. The emergence of Sturm-Liouville theory'' by Lützen).

Actually, problems related to the SL equations are simple to write, because Sturm-Liouville operators only involve differential operators), yet their resolutions involve a great deal of ideas that have been generalized and extended in many directions. For me, dealing with SL problems could be a "smooth way" to learn and understand key ideas without relying on abstract settings. For example, I was recently very happy to read the elegant proof of Weyl on the limit circle and point circle, and to have a glimpse of its relations with the "resolution of identity", as it really enlightens it.