According to Wikipedia and other sources, Tait's conjecture would have had significant effect on solving the FCP had it been true.
Are there other mathematical connections between FCP and hamiltonicity?
MR0975998 (90d:05156) Fleischner, H. The prism of a 2-connected, planar, cubic graph is Hamiltonian (a proof independent of the four colour theorem). Graph theory in memory of G. A. Dirac (Sandbjerg, 1985), 141–170, Ann. Discrete Math., 41, North-Holland, Amsterdam, 1989.
The review says, in part,
Using the 4CT, the result of the title was shown by M. Rosenfeld and D. W. Barnette [Discrete Math. 5 (1973), 389–394; MR0321806 (48 #173)]. Now the author offers a much longer and more intricate proof, which, however, is 4CT-free.
[So, the connection here is that the first proof of Hamiltonicity relied on the Four Color Theorem, but the newer proof didn't.]