In general, if $G$ is a profinite group and $M$ a discrete $G$--module, then $H^i(G,M)$ is torsion for $i>0$. This applies in particular to Galois cohomology, i.e. when $G$ is a Galois group.

Tate modules are not discrete Galois modules, and their cohomology will usually not be torsion. The same goes for $\mathrm{End}(T_\ell A)$.

Over finite or local fields the cohomology of $T_\ell A$ is more or less well understood. Not so over global fields.

In general, if $G$ is a profinite group and $M$ a continuous discrete $G$--module, then $H^i(G,M)$ is torsion for $i>0$. This applies in particular to Galois cohomology, i.e. when $G$ is a Galois group.

Tate modules are not discrete Galois modules, and their cohomology will usually not be torsion. The same goes for $\mathrm{End}(T_\ell A)$.

Over finite or local fields the cohomology of $T_\ell A$ is more or less well understood. Not so over global fields.