What is the difference between the 23 bi-affine and the 39 fully quadratic equations of the rijndael sbox?
Looking at appendix A of the eprint I'd say that the authors call a polynomial of algebraic degree at most $2$ bi-affine, if it contains some terms of algebraic degree $1$, and fully quadratic, if there are no such terms.

(Working over fields of characteristic $2$ the map $x\mapsto x^2$ is linear over the field with two elements and of algebraic degree $1$.)