Turing reducibility is strictly weaker than computable many-one reducibility

With oracle access to $A_{TM}$, an input $x$ belongs to $\overline{A_{TM}}$ exactly when the oracle answers "no" to the single query $x$. So the machine that asks that one query and flips the answer halts on every input and decides $\overline{A_{TM}}$. Hence $\overline{A_{TM}}{\le }_T{A}_{TM}$.

Suppose instead that $\overline{A_{TM}}{\le }_m{A}_{TM}$. Then $\overline{A_{TM}}$ would be recognizable because $A_{TM}$ is recognizable by [L1] and recognizability transfers backward along many-one reductions. Since [L1] also makes $A_{TM}$ recognizable, [L2] would imply that $A_{TM}$ is decidable, contradicting [L3]. Therefore $\overline{A_{TM}}\mathrm{̸}{\le }_m{A}_{TM}$.

Steps 1.1 and 1.2 exhibit a Turing reduction that is not a many-one reduction, so Turing reducibility is strictly weaker than computable many-one reducibility.