This is a very good question and I think it is one of the most common questions that I hear when the topic comes up.
There are essentially two differences:
1) Natural Family Planning (NFP) never breaks the integrity of sexual intercourse.
2) Natural Family Planning (NFP) is ascetic and requires discipline and sacrifice.
The first difference is that sexual intercourse constitutes a single act between a husband and wife. A woman's anatomy and a man's anatomy are constructed for this act and are ordered toward this certain purpose and end. The essential Catholic teaching is that the husband must always ejaculate within his wife's womb without obstruction. This is what constitutes the sexual embrace because it open to the natural purpose of conception. I hope I'm not being too crude, but that is bare truth of it all.
Homosexuality is wrong for the same purpose that contraception is wrong. Both seek the pleasure of sex but do so by divorcing the anatomical and procreative "structure" of sexual intercourse.
NFP also preserves the integrity of the sexual embrace every single time. That's the glaring difference. NFP is basically just practicing abstinence for a certain time of the month when the wife is fertile. This time is discovered by two indicators: temperature (taken orally) and vaginal mucus. The old "rhythm method" was essentially watching the calendar based on previous menstrual cycles. The problem with it is that when a woman is ill or stressed (or any other reason) her cycle can lengthen or shorten). NFP on the other hand analyzes the signs of fertility daily so that you're always on top of the situation. It is therefore very accurate and basically full-proof.
The Catholic teaching is that it's all or nothing. The married couple must preserve the integrity of the sexual act (i.e. ejaculation in the wife's womb without obstruction) every single time. However, married couples are not required to have relations every single day. They can pick their days and if they want to not make love on days of fertility, they are free to do so. They are not compelled to make love during fertile times.
Which brings us to the second difference: asceticism.
NFP requires discipline. A husband can't just have his way with his wife or vice versa at the drop of a hat. They have to practice discipline. Marriage, sex, and procreation are sacred institutions that remain interjoined. You can't just start introducing devices, chemicals, or surgery so that you can have fun. Another way of looking at is like this: If you don't want to be fat, eat less food. It's not appropriate to engage in unnatural quick-fixes like bolemia or anorexia. Being responsible for your body (a holy thing) requires discipline or ascesis. You must eat less or work out. You must deny yourself, something Americans don't want to hear. The same goes for the sacred liturgy of love. If you are not ready for a child, don't make love all the time. Observe the signs and seasons of your God-given fertility and act accordingly.
I hope this is helpful. I imagine that there will be more questions so fire away.