Both results are correct. When Fulton and Harris classify $\text{SL}_n(\mathbb{C})$, they treat $\text{SL}_n(\mathbb{C})$ as a complex Lie group and representations have to be holomorphic, so complex conjugate representations are not allowed. In Tung's, however, he treats $\text{SL}_n(\mathbb{C})$ as a real Lie group, so representations of $\text{SL}_n(\mathbb{C})$ just need to be smooth and complex conjugate representations are permissible.