A bit of a spoiler here: this mitzvah prohibits serving idols according to our practice of worship, while the next prohibits serving them in their own unique manners of worship. (Sit tight, we’ll come to that shortly.)
The point of the verse is not to bow down with the intention of worship (as opposed to, say, bowing down to pick up your bag of oranges that spilled on the ground). This is a form of worship that we use for God; similarly, all such forms of veneration are prohibited for anything other than God Himself. These include slaughtering a sacrifice, burning an offering and pouring libations (see Talmud Sanhedrin 60b).
It should be noted that if you do happen to spill your bag of oranges in front of an idol, you should just let them go. Even if you bent down to pick them up without any intention of worshipping the idol (God forbid!), others might mistake what it is you’re doing. (See Talmud Avodah Zarah 12a.) Remember the story of Chana and her seven sons, during the time of the Maccabees? Even though his six older brothers had already been put to death for refusing to bow to an idol, the youngest refused to relent. The tyrant offered the boy a deal: if he just bent down to pick up a ring in front of the idol, the youngest brother would be released. Nevertheless, the youngest son refused and a Heavenly voice praised the woman for the holiness of her children. (This story is found in Talmud Gittin 57b and also in the Apocryphal book of II Maccabees.)
The prohibition against bowing to idols (and similar forms of service) is discussed in Talmud Sanhedrin (62a) and throughout the tractate of Avodah Zarah. It is codified in the Mishneh Torah in Hilchos Avodas Kochavim and in the Shulchan Aruch in Yoreh Deah 150. It is #5 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #11 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar. (See the next mitzvah for the position of Nachmanides.) This mitzvah is binding upon both men and women in all times and in all places.