PUSH is just an alias for CONS, useful for emphasising that you're working with a stack.

The free list holds all the cons cells not in use. Note that in this machine, there is no way to create or destroy cons cells - that's the whole point! Rather, when you need to cons some values, you pull a cell off the free list and populate it (see the implementation of CONS), and when you want to throw a cons cell away, you put it back on the free list (see the implementations of POP and FREE).

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)

For the former, "push x list" is to "cons x list" as "var += 7" is to "var + 7". It's just syntactic sugar to modify a variable.