This is not specific to Hamiltonian path and circuit. A circuit is a path where the first and last vertices are the same. Another term for a circuit is a cycle.
well that true for Euler circuit, and Euler path, cause Euler path is when you go through all the edges, and circuit has to return to the original vertex, but for hamilton it has to visit all the vertex, but how can it return to original vertex, cause it not allowed to visit one vertex more than once.
There is some disagreement one this point among different authors.
Some simply say: A graph is Hamiltonian if there is a cycle including every vertex exactly once (except for the first and last.
This in effect says that the terms Hamiltonian cycle and Hamiltonian circuit are synonymous.