I will try to answer the question entirely from a Catholic perspective, as that seems to be what the question demands.
The classic determination of a moral act has three parts (CCC 1749-1761):
- Is the act intrinsically immoral --- which is to say, wrong in every circumstance --- by its very nature?
- Is the intention good, or bad?
- Would the circumstances surrounding the moral act make it have an evil result?
A moral act (which is to say, an act in which a human has the freedom to act one way or another) must pass those 3 tests above to make it a moral action.
Now, let's compare non-abortifacient artificial contraception and Natural Family Planning in regards to the tests above. I add the proviso that the artificial contraception be non-abortifacient because otherwise we are talking about abortion (even if early abortion), and such act is fairly equated to murder of the innocent (CCC 2270-2275): an intrinsic evil of its own. It goes without saying that we are assuming any sexual intercourse takes place within marriage in both cases, or else we are talking about fornication, which immediately fails the third requirement (CCC 2353).
In what follows, see (CCC 2366-2370)
Non-abortifacient artificial contraception fails the first test. By its very nature artificial contraception (even when it doesn't kill the unborn) is a contradictory act. On the one hand, the sexual act is supposed to be (among other things) a total self-giving of spouses to each other, and on the other hand, the contraceptive symbolism (and intended effect) is to withhold from each other an integral part of themselves: their fecundity --- whether abundant or slim. In a sense it is very much like deception. One would have strong objections to a man pretending to be someone else in order to marry a woman (and vice-verse) --- it would go against the very nature of marriage that requires honesty and trust.
On the other hand, Natural Family Planning (refraining from having sexual intimacy at times) does not fail this test. There is no moral requirement that a couple have sex every waking moment of their lives. And in fact, there are circumstances when they must not have sexual intimacy (for instance, after surgery in their nether regions). And engaging in Natural Family Planning does not engage in the double-speak that contraceptive sex does: it is the difference between lying and remaining silent (See esp. CCC 2370). While we are called to always tell the truth, we are not always called to speak.
Of course, we are sometimes called to speak, and tell the truth. Analogously, God calls us to "be fruitful and multiply". Here is where we talk about intentions (the second requirement), which gets at the heart of the question. After all, aren't both methods (non-abortifacient artificial contraception and Natural Family Planning) intending to avoid children? And isn't that against God's will for humans stamped in their reproductive nature --- whichever method you use?
To be sure, there is much truth in these questions. If there was not the positive call from God to reproduce, one might be able to respond that both methods have the same ends, but one uses intrinsically flawed means of accomplishment, while the other uses entirely natural means. That difference is not trivial, of course, and would suffice as an answer if there was not the positive command from God to "be fruitful and multiply". However, we are called to reproduce. For now, let me say this: if someone's use of Natural Family Planning has as its goal the avoidance of children for the sake of not having any children, then that person is clearly doing violence to one of the purposes of matrimony (which is to obey the command to "multiply"), and as such has failed the second requirement of a moral act by having the wrong intention. Which is to say, Natural Family Planning is not "intrinsically moral" or "intrinsically good" (always right).
But how can it be right at all? Isn't the command to be "fruitful and multiply" definitive and absolute? Shouldn't the mandate be to find the best time for procreation every month and make sure to have sex during those times (whether or not on other times as well)?
Well, but we have been given other mandates from God on the subject of marriage. One of them is, for example, to care well for the children we do have. In other words, the command of God to "be fruitful and multiply" would not be fulfilled very well at all if we left our children to fend for themselves once we gave them birth. Part of the command is that we must do our best to make sure our children have a good chance of surviving in this world. And that may mean that we must have less children than we are physically able to have in order to be able to feed and care for the ones we do have.
Among other things, we are also called to respect our spouses bodies, for instance, and we must not engage in sexual intimacy when our wife is likely to be fecund if she is physically unable to handle the pregnancy.
In other words, we must be responsible with the gift of fecundity we are given (CCC 2367-2368), and take into account all other commandments from God. But we must have the right intentions. An avoidance of children must be motivated not by the wish not to have to sacrifice --- we are all called to sacrifice, just as Jesus sacrificed Himself --- our affluent lives, or by the wish to make all procreation impossible, or by selfish reasons, but must instead be "in conformity with the generosity appropriate to responsible parenthood" (CCC 2370). Our parenthood must be both responsible and generous, and never selfish. To take one instance, one must not forgo having a child because one really wants a boat.
Okay, but isn't it possible to have this lofty (and very worthy) aims in mind when one uses non-abortifacient artificial contraception to avoid children? Yes! In regards to intentions, the intention to avoid children is the same whether one uses non-abortifacient artificial contraception or Natural Family Planing, and in both cases the intentions can be worthy and moral, or immoral. However, assuming morally acceptable intentions, the question of means now shows up again. Whereas it is quite natural to use Natural Family Planning to space one's children, non-abortifacient artificial contraception is intrinsically evil because, by its very nature, it lies in a very fundamental way which is entirely contradictory to the essence of matrimony.
Finally, circumstances. Some actions which may be otherwise moral, may end up becoming immoral depending on the circumstances. A classic example of this might be the following: while it is morally good to attend Church on Sundays, it might not be moral to do so while in a foreign country that actively persecutes Christians if by doing so the attention of the authorities would be called to an underground group of Christians in hiding. In other words, consequences of an action are evaluated here. I have already included some language of circumstances above in my discussion of intentions. As far as Natural Family Planning goes, its morality might change or might be different if talking about a well-to-do family compared to a poor family, or if talking about a young and healthy couple compared to an elderly or not so healthy couple, or if talking about a large family compared to a small family. The devil here is in the details (and the moralist). In any case, the circumstances do not ever change the nature of an intrinsically evil act to a morally acceptable one (CCC 1754). They might lessen the moral culpability of an evil act, but the action remains as evil as ever. They might also increase the moral culpability of an evil act (lying about one's finances is less problematic when doing it to your friend than to the court, for instance). Similarly, circumstances can also increase or diminish the moral worthiness of a moral act (volunteering for the army would be more courageous in times of war than in times of peace, for instance). In any case, there are no circumstances in which non-abortifacient artificial contraception becomes a good act (that is the meaning of intrinsic evil, after all).
Well, that was too long. I wonder how many will read it all the way through?