In Saunders Mac Lane's Categories for the working mathematician one can read, when talking about fully faithful functors:
[...], but this need not mean that the functor itself is an isomorphism of categories, for there may be objects of B not in the image of T
Given a fully faithful functor $T: C \to B$ and an object together with its identity arrow $c \in C, 1_c: c \to c$, and given $T 1_c = 1_{Tc}: T c \to T c$ in $B$, how can there be a fully faithful functor between categories with unequal amounts of elements?
Because faithful functors follow the rule $Tf_1 = Tf_2 \Rightarrow f_1 = f_2$ and with categories with unequal (or rather less) elements there must be some $1_{Tc}$ that is equal to some $1_{Tc'}$
For me it seems to be the same reason why there is no injective function $\mathbb{Z} \to \mathbb{N}$
Let $\mathcal C$ be any full subcategory of a category $\mathcal D$. Then the inclusion $\mathcal C \subseteq \mathcal D$ gives a functor $F\colon\mathcal C \to \mathcal D$. This functor is always fully faithfull, but it is not always true that a subcategory is isomorphic to the larger category.
For a really down to earth example take the category whose only object is the zero set with its identity map. This includes into the category of all sets and these categories are not isomorphic.