It's the same with every abrahamic religion. Who's the most rabid follower competition.
It's a bit more complex than that. For the initial decades after the state of Israel was established the Ashkenazi were the dominant elite lording it over the Sephardic , the Mizrahim & the other Jewish communities who were miniscule. In fact the overall consensus among the elite was that the Mizrahim were not only backward & therefore an embarrassment but they faced racism & marginalization by the dominant elite which culminated in quite a few riots.
It's only in the 1970s that as their population grew , they set up their own political parties which in turn got into a coalition with Likud headed by a politically rehabilitated Menachem Begin who then became the PM breaking the monopoly over power by the Labour & the Ashkenazi.
Underneath the surface there's still a lot of tension between these groupings . The Ashkenazi is more likely to settle in the west & hold dual citizenship unlike the Sephardic or the Mizrahim who live on in Israel ready to face hardships there. That's one among plenty of other sticking points in their adversarial relationship with each other. Moreover the TFR of the Ashkenazi is rumoured to be around replacement level compared to the Mizrahim who've a TFR of anywhere between 3.0-3.50 if not more. And the latter are among the more conservative / orthodox Jews in Israel.
More than anything else , that should be an indicative of the future of Israel - its policies , its politics , etc.