SURF feature extractor and descriptor
First of all, SURF only uses it as a blob detector to find interest points.
The first order derivatives only give you directional information such as edge-ness. However, the Hessian (as the matrix of second derivatives) contains local structure information such as curvature / concavity etc. For instance, its eigenvectors can be used for cornerness measures. It is just more informative. Also, its determinant is used as an automatic scale selector. 

Another common alternative would be a Laplacian detector, but Hessian shows better scale selection properties.