Humanae Vitae stands or falls on the claim that human sexual activity has a double significance. Not goal or natural end or teleology, mind you, but significance. Paul bases the significance in part on teleological considerations, but he unmistakably wants to speak about more than ends. So what is the difference?
1.) The possibility of exceptions. To say that a thing has a natural order does not suffice to show that it cannot be used for other goals. Screwdrivers, duct tape, coat hangers etc. all have a clear teleology – all are named after from the final cause – but all have a wide application outside of that. Not all natural ends are like this, and Edward Feser has done a very good job showing why the natural end of sexual activity does not have the same wide application outside its natural end, but Paul is coming to the same conclusion by simpler means. There aren’t exceptions to meaning, as though we sometimes use the word “cat” to refer to something that barks. This is either to lie or to change the meaning, both of which contradict Paul’s claim.
2.) Meanings can be equal and multiple. To speak of multiple ends means either that one is subordinate to another or that both are subordinate to some further end. But things can have multiple layers of meaning without having a subordination to each other (like two spiritual meanings of Scripture, or an intentional double meaning in the literal sense).
3.) Meanings are essentially interpersonal, and to act against them is always evil. This is in some ways a support for (1) but it is broader. Sexuality, like language, has an order to manifesting the truth of the person, and this can only be done by respecting the intrinsic meanings as we find them.
4.) Meanings have their end as actually present in the act of signifying, but natural ends need not be actually present in their acts. There is therefore something essentially fruitful in sexual union even where a child is not conceived. Actual conception is not merely separable from this, but there is a real completion and fecundity apart from this.