The definitions coincide, with some caveats: basically for a curve, there is a single notion of genus, which applies equally to smooth curves over algebraically closed fields, and to their function fields; and also over the complex numbers to the associated Riemann surface as two-dimensional manifold. See http://en.wikipedia.org/wiki/Genus_%28mathematics%29 . On the other hand care is needed for curves that are allowed to be singular, or fields that are not algebraically closed, what definition is in use.