The spelling of the word 'admit' has a ⟨d⟩ in both English and the Spanish equivalent, admitir, but not in Italian ammettere. Why is the ⟨d⟩ absent in the Italian equivalent?
admitir is not the inherited form of Latin admittere in Spanish (i.e. it has not undergone the normal Latin > Spanish sound changes), but a later restoration based on the Latin form. If admittere had come down from Latin to Spanish without restoration, it would probably look something like "ameter". (Compare Sp. atenerse "to adjust (to something)" < Latin attinere se.)
By contrast, Italian ammettere is either inherited from admittere, or was restored and then adapted to the expected Latin > Italian sound changes.
Many other common Spanish words (perhaps more than in Italian, overall) are restorations from Latin, and they often co-exist with a more obscure word that has the same origin, but has developed through the Latin > Spanish sound changes: for example, rápido (raudo), regla (reja), quieto (quedo), etc.