Let $M$ be of a type $III_{\lambda}$ factor with separable predual.
If $\lambda=1$, given two normal states $\tau_1$ and $\tau_2$ on $M$.For any $\epsilon>0$, there exists a unitary $u\in M$ such that $\|\tau_1–u\tau_2 u^*\|<\epsilon$.
If $\lambda\neq 1$, does there exist similiar conclusions?
The paper you are reading is devoted to prove that in a III$_\lambda$-factor $M$ $$ \operatorname{diam}(S_0(M)/\operatorname{Int}(M))=2\,\frac{1-\lambda^{1/2}}{1+\lambda^{1/2}}. $$ This shows that what you want can only happen in a type III$_1$-factor.
Edit: why faithful states are enough. Let $\phi,\psi$ be states, with supports $p$ and $q$. Because we are in a type III factors, $p$ and $q$ are equivalent; in fact, unitarily equivalent. So $q=vpv^*$ for some unitary $v$. Then the state $v\psi v^*$ has support $p$ and is equivalent to $\psi$. This reduces the problem to states with the same support $p$. And these are faithful in $pMp$.