What is the difference between a dual cipher and a tweak?
For distinct tweaks $t \ne \tau$, the ciphers $E_{k,t}$ and $E_{k,\tau}$ should appear to be independent uniform random permutations for uniform random $k$.  For example, you could think of it like having a single-bit tweakable block cipher where $E_{k,0} = \operatorname{AES256}_{k_0}$ and $E_{k,1} = \operatorname{Serpent256}_{k_1}$, where $k_0$ and $k_1$ are like magic independent uniform random bit strings somehow both deterministically derived from $k$.

For dual ciphers $E \ne E'$, $E_k$ and $E'_k$ are very much not independent—there is a deterministic relation between them for every $k$, namely $f(E_k(p)) = E'_{g(k)}(h(p))$ for some deterministic functions $f$, $g$, and $h$.