In dimension 5 and higher, there are simply connected rational homology spheres that are not spheres, e.g. the Wu manifold $SU(3)/SO(3)$, see Theorem 6.7 in [2] and Remark, p. 374 in [1]. See also [3] and Lemma 1.1 in [1] for more examples.
[1] D. Barden, Simply connected five-manifolds. Ann. of Math. 82 (1965), 365-385.
[2] M. Mimura, H. Toda, Topology of Lie groups. I, II. Translated from the 1978 Japanese edition by the authors. Translations of Mathematical Monographs, 91. American Mathematical Society, Providence, RI, 1991.
[3] Ruberman, D. Null-homotopic embedded spheres of codimension one. In: Tight and taut submanifolds (Berkeley, CA, 1994), volume 32 of Math. Sci. Res. Inst. Publ., pp. 229-232. Cambridge Univ. Press, Cambridge, 1997.