Classes of graphs with easy Hamiltonian cycle but NP-hard TSP
Cographs are not always Hamiltonian, have polynomial time tests for Hamiltonicity, and are NP-hard to solve the traveling salesman problem for.

More generally the Hamiltonian cycle problem can be solved in polynomial time (but is not fixed-parameter tractible) on graphs of bounded clique-width; see, e.g., Fomin et al., "Clique-width: on the price of generality", SODA'09. But again because these graph families include the complete graphs, the TSP is hard on these graphs.