Why are some wines kosher and some not?
The Torah prohibits wine offered as a libation to idols (Shemos 34:15). Based on a generalization that non-Jews are devout practitioners of their religion, wine made, touched, poured, or tapped by someone who is not an observant Jew was prohibited by the Sages (Shulchan Aruch, Yoreh Deah 123:1, see also Chochmas Adam 76:1) out of concern that it might have been offered to an idol. (Incidentally, in an unrelated discussion a certain work I read mentioned that the Greeks, whose culture was highly influential during the time when the Sanhedrin formalized many Rabbinical laws, held wine libations in particularly high esteem). Additionally the restriction was meant to restrain social interaction somewhat to lessen the possibility of inter-faith marriage (Taz 123:1, Shach 123:1, based on the Gemara)
Wine which has been cooked loses quality and is no longer fit for libation. As such the restrictions no longer apply once wine has been cooked (mevushal) provided it was still kosher up until that point (Shulchan Aruch, Yoreh Deah 123:3).
The question and the answers mostly deal with the spiritual aspects of wine. I am not disagreeing with them. But, it should also be remembered that although wine is a vegetarian product, it still must be prepared in accordance with the laws of kashrus. In other words, all equipment and any additives need to be in accordance with kashrus. The equipment used must be kosher, just like any other food preparation equipment. Some wineries might use products that are not kosher in finishing and aging. Kosher wine will only be made using additives that are also kosher and when a kosher certification is present, we know that the wine was produced as a food product in accordance with laws of kashrus.
If the grapes are from Israel, the wine must have truma and maaser taken from it. If truma has not been taken from Israeli wine, it may or may not have the punishment of spiritual cutting off, depending on separate arguments. Thus perhaps drinking some non-kosher wines is worse than eating pig.
For all wines there is a rabbinical decree that a non-Jew cannot touch it. If it is cooked there is no decree on such wine, since it was not common to drink cooked wine. What is called cooked is of course a disagreement, and the OU has 3 different mevushal heckshers depending on how the wine was cooked. Also whether the wine or grape juice was cooked is called mevushal is another disagreement, which may or may not be a problem for Herzog wines. For wine made for idol worship which excludes most/all wineries today see Yirmeahus answer.
If a Jew breaks Shabbat in public, he has the halachic status of a non-Jew. What is called breaking Shabbat in public and where it does and does not apply is a separate argument.
On one hand, I heard that some interpret the rabbinic decree as being that a non-Jew cannot even look at wine, or it becomes yayin nesech. That is a stringent opinion, which is supposedly observed by certain chassidic communities.
One the other hand, I heard the following leniency in the name of Rabbi Yaakov Etlinger: even if someone is not a sabbath-observer, it doesn't necessarily mean that he denies that Hashem created the world in six days and rested on the seventh. I believe that the Hebrew term for this is mumar leteiavon. Most people nowadays don't actually believe that there are other gods, so according to him, wine cannot acquire the status of "idolatrous libation wine," which really is quite a chiddush. You might have atheists nowadays, but they don't pour libations to gods that aren't.
(Similarly, this is why Rav Kook says that atheism is one step closer to Judaism than pantheism. Rabbi Chaim ibn Attar disagrees based on the verse (Deut. 6:4) Hear, O Israel: the LORD our God, the LORD is one, because first we should consider it as if there are other gods and that Hashem is our God, and then we reject that there are other gods and that Hashem is the only God, Hashem Echad).