Why do ODEs that have solutions that have closed form solutions, *have* closed form solutions? Why do certain classes of ODEs have closed form solutions? Is there something these classes have common apart from the fact that they have closed form solutions?
When I say "closed form", I mean something vague, like there is no finite combination of usual operators (addition, multiplication, powers, trigonometric and logarithmic operators) and operands that can be used to write the solution. 
 A: Determining whether series have "closed form" solutions is an incredibly difficult task.
For non-linear DEs this is done using "symmetry" techniques (as mentioned cryptically in the comments). Indeed Peter Olver's text is a good one for delving into this stuff. Other introductory texts are "Symmetry Methods for Differential Equations: A Beginner's Guide" by  Peter E. Hydon (this is a much lower level text than Olvers). Also, "Introduction to Symmetry Analysis" by Brian Cantwell. Cantwell's text has a wealth of information but is written to more of an engineering crowd. 
The question of determining whether you can "solve" a differential equation, led to the development of a large branch of modern mathematics known as "Lie Theory" (named after Sophus Lie - pronounced "LEE"). Lie wanted to do for differential equations what Galois had done for polynomial equations.
Now if you restrict yourself to (systems) of linear differential equations, you can answer solvability by computing "differential Galois groups". Here the texts by Andy Magid ("Lectures on Differential Galois Theory") or Singer/van der Put ("Galois Theory of Linear Differential Equations") are good references. Although Crespo's "Algebraic Groups and Differential Galois Theory" would be where I started.
Here one looks are "symmetries" of differential field extensions and get an analogy of Galois theory for regular field extensions. If the differential Galois group of a differential field extension is "solvable", then the system of DEs is "solvable" in a very precise sense.
In Magid's book you can find proofs of things like: $\int e^{x^2}\,dx$ isn't an elementary function and the the Airy equation ($y''-xy=0$) is "unsolvable" (one can find series solutions but not your sort of "closed form" solutions).
All that said, whether there are "closed form" solutions or not is a very deep and difficult question to ask. It's even hard to formalize what you should mean by "closed form". :)
