Graph classes where Hamiltonian Cycle and Hamiltonian Path problems have different complexity
The Hamiltonian path problem on grid graphs with maximum degree 3 is NP-complete. The proof is in C. H. Papadimitriou and U. V. Vazirani, On two geometric problems related to the travelling salesman problem, Journal of Algorithms, Volume 5, Issue 2, June 1984, Pages 231–246 (Theorem 2)