If functors are morphisms between categories, and natural transformations are morphisms between functors, what's a morphism between natural transformations? Is there ever a need for such a notion?
(Small) categories form what's called a 2category, which is a structure that has objects, morphisms (functors), and morphisms between morphisms (natural transformations). There are also ncategories, which have a deeper morphisms structure. A google search will point you to a lot of references about ncategories. But for ordinary categories, the story ends at natural transformations. 


The next one is called Modifications between ordinary natural transformations between functors between categories are trivial, but they exist between between 2functors between 2categories. The next one after that is called perturbation : a perturbation goes between modifications between pseudonatural transformations between 3functors between 3categories. Beyond that, no established terms exists. Instead one starts numbering things and speaks of nFunctors are 0transfors. Transformations are 1transfors. Modifications are 2transfors. Perturbations are 3transfors. And so on. 


One advantage of the abstraction of category theory is that one is not constrained to "concrete" objects and morphisms (I mean, made by set with a structure together with functions preserving it), and constructions of new categories from simpler ones are very easily performed. As a result, any further and more general categorical notion can always be read as a particular case of a simpler and more basic one, in a suitable category. Thus in the proper context, a morphism is an object; a natural transformation is a morphism; similarly, a universal arrow is a particular case of an initial object, which of course is a particular case of a universal arrow, and so on. So in a sense, there is no need of the notion of "morphism between natural transformations", just because it is already a particular case of a more basic notion already defined. In practice, several used categories (e.g. algebras; preshaves; chain complexes,...) are themselves categories of functors, where arrows are natural transformations. In this context a morphism between natural transformation naturally arises, even if it will be seen as just an ordinary morphism. 


This is not really a sophisticated answer as the other ones, but maybe it makes visible why higher structures are needed to get an interesting notion of morphism between natural transformations. Assume $F,G : C \to D$ are functors and $\alpha, \beta$ are natural transformations $F \to G$. What could be a morphism $\alpha \to \beta$? Since $\alpha$ and $\beta$ consist of their components $\alpha(c) : F(c) \to G(c), \beta(c) : F(c) \to G(c)$, the only reasonable way of "connecting" these data in our category $D$ is by means of two morphisms $\gamma(c) : F(c) \to F(c), \delta(c) : G(c) \to G(c)$, so that the resulting diagram becomes commutative. Furthermore, $\gamma$ and $\delta$ should become natural transformations $F \to F, G \to G$. But this comes from a more general concept, namely the arrow category: If $C$ is a category (in the above case, this is a functor category), its arrow category has as objects the morphisms of $C$ and a morphism between two morphisms $x \to y, x' \to y'$ is a pair of morphisms $x \to x', y \to y'$, making the obvious diagram commutative. This may be also described as the functor category $C^{\textbf{2}}$. 

