A guided ion beam tandem mass spectrometer is used to examine the kinetic energy dependence of reactions of the third-row transition metal cations, Ta+, and W+, with molecular hydrogen and its isotopologs. A flow tube ion source produces Ta+ and W+ ions in their electronic ground state term and primarily in the lowest spin-orbit level. Corresponding state-specific reaction cross sections are obtained. Modeling of the endothermic reaction cross sections yields the 0 K bond dissociation energies in eV (kJ/mol) of D-0(Ta+-H)=2.38+/-0.06 (230+/-6) and D-0(W+-H)=2.27+/-0.05 (219+/-5). The experimental thermochemistry is consistent with ab initio calculations, performed here and from the literature, which also provide the electronic structures of these species and details about the reaction surfaces. Results from reactions with HD provide insight into the reaction mechanisms and indicate that these early metal ions, Ta+ and W+, react largely via insertion mechanisms. Results for these third-row transition metal systems are compared with the first-row and second-row congeners and found to have higher reactivity towards dihydrogen and stronger M+-H bonds. These differences can be attributed to the lanthanide contraction, relativistic effects, and efficient spin-orbit interactions among surfaces of different spin.