The question here is « If both Judaism and Islam teach love, why do the two sides harbor so much hate? » and this question is argumented by the different holy texts saying to care for your neighbour. Now there are 4 things I would like to mention (correction is valued if my response is incomplete):
1- You need to contextualise the texts. Maybe the treatment is different on your neighbour if his religion is different from yours. Maybe if there’s a wall separating each other, one cannot simply care (as in love) for the other. That’s why context is necessary (and I won’t get into it as I don’t deem it needful for now).
2- Whilst the holy texts do say this, they do also contain texts on how to combat your enemy (which the Bible has as well). One cannot simply stand there while injustice is done in the hopes of the « mischiveous » to be guided and forgiven.
3- And also you did admit that the question did sound « kumbayaesque », which is true and I thank you for your honesty. As you probably can tell, a conflict that has endured for so many decades (which has begun even before Israel’s creation) is more or less likely to have a fortune goal. We, and I hope the « Pro-Israel Coalition » as well, do hope that this madness will soon stop.
4- And fourth, not everyone really follows the sacred books. Suicide-bombing is forbidden in Islam, as is bombing hospitals in Judaism (if they consider themselves Jews). There’s proper religious ways to fight your enemy.