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We study the contact equivalence problem for toric contact structures on$S^3$-bundles over $S^2$. That is, given two toric contact structures, one canask the question: when are they equivalent as contact structures whileinequivalent as toric contact structures? In general this appears to be adifficult problem. To find inequivalent toric contact structures that arecontact equivalent, we show that the corresponding 3-tori belong to distinctconjugacy classes in the contactomorphism group. To show that two toric contactstructures with the same first Chern class are contact inequivalent, we useMorse-Bott contact homology. We treat a subclass of contact structures whichinclude the Sasaki-Einstein contact structures $Y^{p,q}$ studied by physicists.In this subcase we give a complete solution to the contact equivalence problemby showing that $Y^{p,q}$ and $Y^{p'q'}$ are inequivalent as contact structuresif and only if $p\neq p'$.

On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$