there is SOME tension between ashkenazi and sephardim but that was mostly back at the begining of Israel. By and large the entertainment industry had a lot to do with erasing that hatred between the two groups, and its an interesting story. If you look at the pop music coming out of Israel at the begining you see (hear) that its all very... (I can't think of a non racist way to say this)... white and boring and clean and sterile. As sephardic influence starting rearing its head into the music scene, that boring music was over and the whole idea of the white european ashkenazi jew as being more civilized etc. was dealt a blow.
Today, the tension only manifests itself in very superficial jokes (someone will make a joke about someone being a pussy and ashkenazi, or being dumb and sephardic), but its just a matter of words at this point, not hate, not actively doing anything, and even that is very un PC. Perhaps you are referring to tension between the ultra religious ashkenazi and the rest of Israel, which has almost nothing to do with them being ashkenazi.