Your confusion is the following: you have two categories $1$ and $\Omega$ which are equivalent, but not isomorphic. You note that for any category $C$ there is a unique functor $C \to 1$ but that functors $C \to \Omega$ seem to correspond to full subcategories.
There is a tight analogy between categories and topological spaces (so tight it can be realized by a functor, the geometric realization of the nerve) that goes like this:
- Categories are like spaces.
- Functors are like maps between spaces.
- Isomorphisms of categories are like homeomorphisms of spaces.
- Natural transformations are like homotopies.
- Functor categories are like spaces of maps between spaces.
- Equivalences of categories are like homotopy equivalences of spaces.
In particular, two spaces (say the point and the interval $[0, 1]$, which are quite analogous to the terminal category and $\Omega$) can be homotopy equivalent, but not homeomorphic, and hence the set of maps from some space $X$ into those two spaces can look very different. However, it's always true that if two spaces are homotopy equivalent, then the spaces of maps from $X$ to those two spaces are also homotopy equivalent: for example, the space of maps from a space $X$ into $[0, 1]$ is always contractible.
The corresponding statement for categories is that if two categories are equivalent, then the categories of functors (and natural transformations) from any other category $C$ to those categories are equivalent. In particular, we conclude that the category of functors $C \to \Omega$ must in fact be equivalent to the terminal category.
So, what are the morphisms in this category anyway? Suppose $F, G : C \to \Omega$ are two functors. What is a natural transformation between them? Well, it's a collection of maps $\eta_c : F(c) \to G(c)$ such that... but wait. In $\Omega$ there's a unique morphism between any two objects. Hence all of the $\eta_c$ are in fact uniquely determined, and automatically satisfy the necessary compatibility condition to define a natural transformation. Moreover, every such natural transformation is invertible because its inverse is also uniquely determined!
At this point let me introduce the following useful terminology: a category is contractible if there is a unique morphism between any two objects. This condition is equivalent to being equivalent to the terminal category. What we've written down is more or less a proof that if $\Omega$ is a contractible category, then so is any functor category $[C, \Omega]$.