Let $P_{n,d}$ respectively $F_{n,d}$ denote the cone of polynomials in $n$ real unknowns of degree $d$ that are positive respectively that can be written as sums of squares of polynomials. The difference of these two cones is important in the theory of sums of squares approaches for polynomial optimization. In this talk I will discuss this question in the setting of symmetric polynomials and discuss the relation of the asymptotical behaviour of these cones to symmetric inequalities. In this context I will proove that in contrast to the non-symmetric case as $n$ tends to infinity there are not substantially more positive polynomials then sums of squares. In particular in the case of symmetric forms of degree 4 the cones coincide asymptotically