The Veronese surface in $\mathbb{P}^5$ is indeed the only secant defective surface that is not a cone. This is a classical (and non-trivial) result by F. Severi, see p. 6 and Theorem 10.1 in

<cite authors="Ciliberto, Ciro; Russo, Francesco">_Ciliberto, Ciro; Russo, Francesco_, [**Varieties with minimal secant degree and linear systems of maximal dimension on surfaces**](http://dx.doi.org/10.1016/j.aim.2004.10.008), Adv. Math. 200, No. 1, 1-50 (2006). [ZBL1086.14043](https://zbmath.org/?q=an:1086.14043).</cite>

ad some of the references cited in the Introduction of the same paper, notably [14, 54, 57].