Influenza vs. ascendente
While both influenza and ascendente translate to influence, they operate in different scopes. Influenza is the broad, standard term for an effect on anything, while ascendente describes a specific psychological power, charisma, or moral authority one person holds over another.
Influenza
B1This is the most common and neutral term used to describe the capacity to have an effect on the character, development, or behavior of someone or something. It can apply to abstract concepts, external factors, politics, health, and social dynamics.
La musica americana ha avuto una forte influenza sulla cultura europea.
(American music has had a strong influence on European culture.)
Non devi lasciare che la sua opinione abbia influenza sulle tue decisioni.
(You must not let his opinion have an influence on your decisions.)
Il clima esercita una notevole influenza sull'agricoltura locale.
(The climate exerts a considerable influence on local agriculture.)
Sotto l'influenza dell'alcol, ha detto cose di cui si è pentito.
(Under the influence of alcohol, he said things he regretted.)
Ascendente
C1This word specifically refers to the power, prestige, or charm that a person exercises over the will or mind of another. It implies a vertical relationship where one person looks up to or is captivated by the other due to charisma or moral authority.
Il nonno ha sempre avuto un grande ascendente sui nipoti.
(The grandfather has always had a great sway over his grandchildren.)
Grazie al suo carisma, esercita un forte ascendente sui suoi colleghi.
(Thanks to her charisma, she exerts a strong hold over her colleagues.)
Quel politico ha perso tutto il suo ascendente sugli elettori.
(That politician has lost all his sway over the voters.)
Lei subisce l'ascendente del suo insegnante e segue ogni suo consiglio.
(She falls under the sway of her teacher and follows his every advice.)
Summary
Use influenza for almost any situation regarding cause and effect, whether it involves people, markets, or weather. Use ascendente only when describing the personal magnetism, prestige, or psychological authority that one individual holds over others.