They are, in fact, equivalent, in the sense that one can be transformed into the other.
Suppose that your data is represented by a vector $\boldsymbol{x}$, of arbitrary dimension, and you built a binary classifier for it, using an affine transformation followed by a softmax:
\begin{equation}
\begin{pmatrix} z_0 \\ z_1 \end{pmatrix} = \begin{pmatrix} \boldsymbol{w}_0^T \\ \boldsymbol{w}_1^T \end{pmatrix}\boldsymbol{x} + \begin{pmatrix} b_0 \\ b_1 \end{pmatrix},
\end{equation}
\begin{equation}
P(C_i | \boldsymbol{x}) = \text{softmax}(z_i)=\frac{e^{z_i}}{e^{z_0}+e^{z_1}}, \, \, i \in \{0,1\}.
\end{equation}
Let's transform it into an equivalent binary classifier that uses a sigmoid instead of the softmax. First of all, we have to decide which is the probability that we want the sigmoid to output (which can be for class $C_0$ or $C_1$). This choice is absolutely arbitrary and so I choose class $C_0$. Then, my classifier will be of the form:
\begin{equation}
z' = \boldsymbol{w}'^T \boldsymbol{x} + b',
\end{equation}
\begin{equation}
P(C_0 | \boldsymbol{x}) = \sigma(z')=\frac{1}{1+e^{-z'}},
\end{equation}
\begin{equation}
P(C_1 | \boldsymbol{x}) = 1-\sigma(z').
\end{equation}
The classifiers are equivalent if the probabilities are the same, so we must impose:
\begin{equation}
\sigma(z') = \text{softmax}(z_0)
\end{equation}
Replacing $z_0$, $z_1$ and $z'$ by their expressions in terms of $\boldsymbol{w}_0,\boldsymbol{w}_1, \boldsymbol{w}', b_0, b_1, b'$ and $\boldsymbol{x}$ and doing some straightforward algebraic manipulation, you may verify that the equality above holds if and only if $\boldsymbol{w}'$ and $b'$ are given by:
\begin{equation}
\boldsymbol{w}' = \boldsymbol{w}_0-\boldsymbol{w}_1,
\end{equation}
\begin{equation}
b' = b_0-b_1.
\end{equation}
