Proving that the halting problem is not Turing-reducible to the acceptance problem for Turing machines
The two problems are equivalent up to many-one reductions.

Reducing $\mathrm{Halt_{TM}}$ to $A_{\mathrm{TM}}$. Given a Turing machine $M$ and an input $w$, let $M'$ be the machine obtained from $M$ by marking all rejecting states as accepting. Then $\langle M,w \rangle \in \mathrm{Halt_{TM}}$ if $\langle M',w \rangle \in A_{\mathrm{TM}}$.

Reducing $A_{\mathrm{TM}}$ to $\mathrm{Halt_{TM}}$. Given a Turing machine $M$ and an input $w$, let $M'$ be the machine obtained from $M$ by getting to an infinite loop at every rejecting state. Then $\langle M,w \rangle \in A_\mathrm{{TM}}$ if $\langle M',w \rangle \in \mathrm{Halt_{TM}}$.