Can somebody give me an example of a subcategory of an abelian category which is also an abelian category, but not an abelian subcategory (which means some kernels or cokernels are different from the initial ones).
Here is a small example of a full subcategory that is abelian, but not an abelian subcategory:
Let $k$ be a field and let $R_n$ denote the ring of upper triangular $n \times n$-matrices over $k$. Then $mod (R_3)$ is an abelian category with $6$ indecomposable objects (up to isomorphism). There is a unique indecomposable object $P$ of length $3$. It has a simple socle $s(P)$ and a simple top $t(P)$. Consider the full additive subcategory with indecomposable objects $P$, $s(P)$, and $t(P)$. This category is equivalent to $mod (R_2)$ and therefore abelian. It is not an abelian subcategory since the lengths do not add up in $mod (R_3)$.