Let D(K) be the positively-clasped untwisted Whitehead double of a knot K, and T(p,q) be the (p,q) torus knot. We show that D(T(2,2m+1)) and D2(T(2,2m+1)) differ in the smooth concordance group of knots for each m>1. In fact, they generate a Z2 summand in the subgroup generated by topologically slice knots. Our main tool is the Heegaard Floer correction term for the double cover of S3 branched along a knot. We also present some sufficient conditions for general knots to have this independence property. Interestingly, these results are not easily shown using other concordance invariants such as tau-invariants of the knot Floer theory and s-invariants of Khovanov homology.