The main difference, in Common Lisp at least, is that push modifies the list to add the new cell. Cons returns a new list with the cell added so the old list remains untouched.

  CL-USER> (let ((list (list 1 2 3 4)))
             (cons 5 list)
             list)
  (1 2 3 4)

  CL-USER> (let ((list (list 1 2 3 4)))
             (push 5 list )
             list)
  (5 1 2 3 4)