Unless I'm mistaken aureate phoenix increases relic boost, hence MM spell is applied to the aureate phoenix boost too. So the total production is given, for T1-T3 production boosted with MM (when both phoenix boost can apply), by BP×(1+RB+AP)×(1+MM+SP) where BP is base unboosted production, RB is relic boost (includes Mountain Halls effect), MM is spell boost (includes Elvrnar Trade Center effect), and AP, SP are boosts for both Phoenixes.
So the % increase from Aureate Phoenix is AP/(1+RB), with AP=3 for level 10 phoenix and 1+RB varying between 8 and 9.84 for max relic boost depending on Mountain Halls level. This ratio varies between 30.5% and 37.5%.
On the other hand, the increase from Storm Phoenix is SP/(1+MM), with SP=0.5 for level 10 and 1+MM varying between 1.5 and 2.06 depending on Elvenar Trade Center level. This ratio varies between 24.3% and 33.3%.
So unless you have high level Mountain Halls and no ETC, Aureate is slightly better than Storm.