The only difference between the two is rescaling the basis vectors, i.e. conjugating by a diagonal matrix.

For instance with the representations of the symmetric group, the usual choice for the seminormal representation would be to have matrices of the form

$$ \begin{pmatrix} -1/k & 1 \\ 1 - 1/k^2 & 1/k \end{pmatrix} $$

for switching between two tableaux by a transposition $s_i$ of two boxes with axial distance $k$ between them. (See for instance question 66602 which attempts to provide motivation for this particular convention.)

On the other hand, the orthogonal representation is what you would expect, giving matrices

$$ \begin{pmatrix} -1/k & \sqrt{1 - 1/k^2} \\ \sqrt{1 - 1/k^2} & 1/k \end{pmatrix}. $$

The only difference between the two is rescaling the basis vectors, i.e. conjugating by a diagonal matrix.

For instance with the representations of the symmetric group, the usual choice for the seminormal representation would be to have matrices of the form

$$ \begin{pmatrix} -1/k & 1 \\ 1 - 1/k^2 & 1/k \end{pmatrix} $$

for switching between two tableaux by a transposition $s_i$ of two boxes with axial distance $k$ between them. (See for instance question 66602 which attempts to provide motivation for this particular convention.)

On the other hand, the orthogonal representation is what you would expect, giving matrices

$$ \begin{pmatrix} -1/k & \sqrt{1 - 1/k^2} \\ \sqrt{1 - 1/k^2} & 1/k \end{pmatrix}. $$