Rabbi Berel Wein has suggested that long ago, there were a certain amount of anti-Sephardic animosity related to the fact that when during the Crusades, the Ashkenazic Jews forced to choose between the cross and the sword went to their deaths; whereas during the Spanish Inquisition, many Spanish (i.e. Sephardic) Jews chose to stay alive and outwardly profess Christianity.
In the cases of schools today, sometimes the school feels extremely rigidly that it should have the text that its students use for prayer, for instance; and it feels that if this is diluted by students who use different texts, then other policies, or the esprit de corps, could be weakened too. (If the Syrian girl can follow her rabbis and pray differently, then what about the girl whose rabbis allow her to wear pants to school?)
Most strongly, though, it's been observed that in Israeli society today, Ashkenazic Jews tend to follow a fairly sharp divide between "Orthodox" and "non-Orthodox." For many Sephardic Jews, there's more of a continuum, and you're more likely to see those who affiliate Orthodox while less observant. I don't recall the article off-hand, but it was pointed out a non-observant Jew in Israel would never dream of visiting his Ashkenazic Haredi relatives unless he was wearing a kippa during the visit; if visiting Sephardic relatives, he could walk in without a kippa. Thus many Sephardic children are more used to the idea of non-observant Jews, and being somewhat okay with them. If you have an educational model that's all-or-nothing that stresses, and thrives on, the "religious/irreligious" divide, then this would be problematic.
Mind you, I'm not defending any of these practices or attitudes! Just helping to explain some ideas behind them.