Maybe a newbie point of view may help.For a quant, I guess closed-form refer to analytical expressions which are more or less difficult to implement in a computer but are faster to evaluate than the underlying experiment they refer to. Like ln2 for instance.Another advantage comes when one tries to modify the hypotheses related to the closed form and cover more general cases or slightly different ones. Then, one way to go is to try to find another closed form, which is a 'perturbation' of the first one, and have criteria for the validity of this expansion.Also, as in many fields, closed form allows for determining which parameters are relevant, and how they drive the closed-form.About the meaning of any closed form, I think in most cases a closed-form brings very little to our understanding. Most often, it is its derivation that contains most of what is useful, and of course, the hypotheses made.Let's give few examples of my point of view :1 - Assume you would like to compute an integral of a very complicated functions over a stochastic process. But because you are clever, you notice that both the process and the integrand have very peculiar symmetries which, once combined, will give you the trivial result : 0. I guess this could be called the ultimate closed-form. It is clearly very useful to anyone to know that form : no one knowing it would run a simulation of corresponding paths of the process and compute the values of the function along them. But if someone wants to modify a little the function, he should check first whether the new function also has the very peculiar symmetries that simplifies the all stuff.2 - Your problem is to find a way to bike over a road with a random profile carrying a given load of stone, but most of all, to find the work that you'll do doing this. Someone else tells you that if the road profile, even random, follows some general rules, then he's able to find a fancy formula for the work. Again, this is very helpful, but your only knowledge will be what comes out of the formula. On the other hand, if you try to derive it yourself, you could come with the observation that the more the road goes up, the greater the chance it will go down after that. At each time step, you'll find also that you'll have to stay on your bike, which will involve some dynamical corrections of your position on it, to balance your weight with the load, the slope of the road that has suddenly change, and your speed. Again, the closed-form formula is useful but not as is the complete understanding to obtain it. [I'm sure the analogy with what I'm thinking to is obvious.]3 - You would like to know how does a ball oscillate in a quartic well. In fact, your well is not completely quartic, it is quadratic, but with a rather small quartic contribution to its curvature. Again, if someone gives you a fancy formula, you're happy with it, provided of course that the guy also told you when you can apply it, e.g when the ratio of quartic over quadratic is small enough. So, if you need that because you spend all your day long at the desk waiting for phone calls like : hey, I have a well of this kind and a ball of that one, what will be the oscillation period? You don't really care about the derivation of your expression. But if you sit in an obscure office quite far from that desk, and ask yourself what you could do to improve this closed-form with, say, cubic terms in the well profile, you must understand how the expression was derived and that, as soon as correction to quadratic are small, the new period will be modified by adding multiple of the first one : T + 2T + 3T ... Therefore, if by chance, you bring some coffee to the very nice guy sitting at the desk the day he receives an unusual phone call about an unusual well profile, you can just tell him : if corrections to quadratic are small, don't take them into account, it will just add harmonics, but the main oscillation will prevail.