Difference between Hamiltonian path and ST

I was reading up algorithms for finding the minimum spanning tree(in case of weighted graphs) and for finding if a graph has a hamiltonian path(which depends on the presence of a hamiltonian cycle). I got everything messed up. So whats the difference between a hamiltonian path and a spanning tree? Both cover all the vertices in the graph. While we can have efficient algorithms to find a spanning tree(perhaps a minimum spanning tree), why cant we have algorithms for finding a hamiltonian circuit?? We can keep on adding and removing one edge at a time till we reach a cycle and perhaps, we could find a hamiltonian cycle??