Imagine, you are a medical doctor on an intensive care unit. You have a patient with a strong fever and a given number of blood cells and a given body weight and a hundred different data and you want to predict, if he or she is going to survive. If yes, he is going to conceal that story about his other kid to his wife, if not, it is important for him do reveal it, while he can.
The doctor can do this prediction based on the data of former patients he had at his unit. Based on his software knowledge, he can predict using either a generalized linear regression (glm) or via a neural net (nn).
1. Generalized Linear Model
There are far to many correlated parameters for the glm so to get to a result, the doctor will have to make assumptions (linearity etc.) and decisions about which parameters are likely to have an influence. The glm will reward him with a t-test of significance for each of his parameters so he might gather strong evidence, that gender and fever have a significant influence, body weight not necessarily so.
2. Neural net
The neural net will swallow and digest all information that there is in the sample of former patients. It will not care, whether predictors are correlated and it will not reveal that much information, on whether the influence of body weight seems to be important only in the sample at hand or in general (at least not at the level of expertise that the doctor has to offer). It will just compute a result.
What method to choose depends on the angle from which you look on the problem: As a patient, I would prefer the neural net which uses all available data for a best guess on what will happen to me without strong and obviously wrong assumptions like linearity. As the doctor, who wants to present some data in a journal, he needs p-values. Medicine is very conservative: they are going to ask for p-values. So the doctor wants to report, that in such a situation, gender has a significant influence. For the patient, that does not matter, just use whatever influence the sample suggests to be most likely.
In this example, the patient wants prediction, the scientist-side of the doctor wants inference. Mostly, when you want to understand a system, then inference is good. If you need to make a decision where you cannot understand the system, prediction will have to suffice.