"How is NFP different from contraception, since both of them are ways not to have a baby? I can't see a real difference, since the intention is the same: to prevent a pregnancy!"
The question assumes that avoiding pregnancy is inherently immoral according to the Catholic Church. But the Church does not teach this! In fact, it is not inherently immoral to avoid pregnancy.
The question is flawed because it forgets the distinction between the end (our intended goal) and the means (how we get to that goal).
For an act to be moral, both the end and the means must be moral.
The "end" in our particular question is "preventing pregnancy". That is the goal or intention, and it is not inherently evil. If serious reasons exist, a married couple is justified in postponing pregnancy, even indefinitely. So, in such a case, we've got a moral "end".
So far, so good!
But now what about the "means"?
Well, the means a couple uses to prevent pregnancy has to be moral, too. Like so many other things in life, we have a choice between moral and immoral ways to get to a good end. NFP is a moral means of preventing pregnancy, contraception is an immoral means of preventing pregnancy.
Well, because contraception fundamentally changes the very nature of sex. In some cases, the spouses' bodies themselves are altered through chemical or surgical means, and in other cases there is a literal, physical barrier put between husband and wife at the moment they are called to be united as "one flesh". Contraception subverts the inherent, purposeful connection between love and life (i.e., sex and procreation), and when those two dimensions of sex are artificially separated, the marital act itself becomes disordered. To deliberately sterilize the marital act is to strip it of its transcendence and meaning. Sex leaves the sacred and becomes common.
With NFP, by contrast, the spouses and the marital act remain untouched and unaltered. Husband and wife unite in love as they are designed, and fertility works within the rhythms that God (and Natural Law) intends. The integrity of the martial act is completely intact.
But wait! you might say. You are still manipulating the process by not having sex during times you might get pregnant!
No, not manipulating. NFP users are working with the natural cycles of fertility and infertility that God Himself put in place. God could have chosen to make us fertile 100% of the time, but in fact He made us infertile the majority of the time. To prayerfully consider our situation and then avail ourselves of the naturally infertile times of a woman's cycle does not disrespect God's design or the Natural Law.
Remember, choosing to abstain from sex is not a sin. However, choosing to take the pleasure of sex while willfully disconnecting it from its full meaning is a sin.
This analogy may help:
Trixie and Pixie both want to lose weight. Losing weight is not inherently sinful, and can be a good goal.
Trixie's means to that end? She practices self-control and sacrifice by eating healthier meals and smaller portions, and even fasting for a time.
Pixie's means to that end? She eats all she wants, in any portion, heavy on the sweets and treats. Each time she is done indulging her palate, she retreats to the bathroom where she sticks her finger down her throat and vomits it all back up.
I hope we all can agree that Trixie's means of losing weight is moral and ordered, while Pixie's means of losing weight is immoral and disordered.
Bulimia contradicts the body's design by accepting the pleasure of eating, but willfully thwarting its life-giving purpose.
Contraception contradicts the body's design by accepting the pleasure of sex, but willfully thwarting its life-giving purpose.
Natural Family Planning is a good means to a good end, thus is ordered and moral.
Contraception? It's just sexual bulimia.