Poder vs Conseguir: Ability and Permission
Poder expresses general ability, possibility, or permission. Conseguir expresses succeeding at a specific attempt — being able to do something on a particular occasion.
PortuStories Editorial
European Portuguese Grammar Guide
Poder expresses: (1) physical or general ability, (2) permission, (3) possibility.
Posso ajudar-te?
Can I help you?
Permission/offer.
Podes vir amanhã?
Can you come tomorrow?
General availability/ability.
Conseguir means to manage to do something, to succeed in a specific attempt. It implies effort or achievement.
Não consegui abrir a porta.
I couldn't manage to open the door.
Specific failed attempt.
Finalmente conseguiu falar com o diretor.
She finally managed to speak to the director.
Both are grammatical, but they say different things: "Não consegui abrir a porta" describes a specific attempt that failed; "Não pude abrir a porta" implies something external prevented you (no key, no permission). Learners tend to over-use poder where a native would say conseguir.
From a PortuStories Story
Tentei várias vezes mas não consegui resolver o problema. Podes ajudar-me?
I tried several times but couldn't manage to solve the problem. Can you help me?