We do celebrate the death of Haman during Purim. How does that differ from the death of others who want to exterminate us if we are not suppose to rejoice in the death of our enemies?
You ask a very important question. We are taught that those who love good and G-d, should not tolerate evil (Tehilim 97, 10). We are not allowed to ‘turn the other cheek”, but are obligated to pray, and even fight and defeat evil (Rambam, M’lachim ch. 4-5), as we see our heroes do throughout the Tanach (Bible). It is immoral towards society and towards the evil person himself (!) to suffer and tolerate his immorality. The Talmud (Megilla 16a, Psachim 117a) recounts that it is natural (i.e. G-dly), to rejoice when our enemies are defeated, and we must even sing songs of thanks to G-d when they are overcome. Although we find some versions of the midrash where G-d rebukes the angels for singing at the downfall of the Egyptian enemies (Yalkut Shimoni, Divrei HaYaminII, 1085), we are man and not angels, and for us it is natural, and desirable to do so. Hizkiya was even punished for not singing praise after the defeat of Sanherev (ibid), and so does Devora (Shoftim 5), and David throughout Tehilim (Psalms). Judaism is not naïve, and we know from 3,300 years of anti-Semitism, that even though we hate fighting, we must be prepared to do so. Morality must be real, not artificial, and it is clearly unnatural not to hate an anti-Semite who hates you. More so, celebration for the fall of the enemies of Israel shows a positive love and national identification, and conversely, only one who isn’t truly worried when Israel goes to war, doesn’t rejoice when we are victorious over our enemies. The Talmud (Megilla 16a) explains that the Bible’s teaching that we are not supposed to rejoice in the death of our enemies (Proverbs 24, 17), which apparently contradicts many places where we do so in the Bible, refers to if the bad person is our brother, a Jew. In that case, we do not pray that he die, but that he should change his evil ways (Brachot 10a), and try and convince him, as well (VaYikra 19, 17). If his actions hurt others, he obviously should be punished and deterred, nevertheless, he is our brother, and as such, receives special patience. On the other hand, Judaism leaves zero tolerance for the Hamans, Hitlers, Ahmadinijans or any anti-Semite or Jew-hater. Not only is such tolerance immoral, but we’re not allowed to rely on miracles! With Love of Israel, Rabbi Ari Shvat