Using a class of permutation polynomials of F_32h+1 obtained from the Ree-Tits slice symplectic spreads in PG(3,3_2h+1),we construct a family of skew Hadamard difference sets in the additive group of F_32h+1.  With the help of a computer,we show that these skew Hadamard difference sets are new when h=2 and h= 3.  We conjecture that they are always new when h > 3.  Furthermore, we present a variation of the classical construction of the twin prime power difference sets, and show that inequivalent skew Hadamard difference sets lead to inequivalent difference sets with twin prime power parameters.