Can a graph have a cycle containing all nodes but not Hamiltonian?
Just pick a tree which - by definition - doesn't have any simple cycle, but is connected, so you can visit all the nodes and return to the starting one but you must pass on some nodes more than once.

There are also graphs which have an Eulerian cycle (a cycle that traverses every edge exactly once) but not an Hamiltonian cycle. For example pick a graph made with two triangles having one vertex in common.