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Ayudar vs. asistir

While both ayudar and asistir can be translated as to help or to assist, they are not always interchangeable. Ayudar is the general, all-purpose word for to help, while asistir is more formal and also has a very common, distinct meaning.

Ayudar

A1
Ayudar is the most common and versatile word for to help. It is used in all contexts, from informal to formal, and implies actively participating or providing support to make something easier for someone else.
¿Me puedes ayudar a cargar estas cajas?
(Can you help me carry these boxes?)
Mi hermana me ayudó con la tarea de matemáticas.
(My sister helped me with the math homework.)
Hablar con un amigo me ayuda a sentirme mejor.
(Talking with a friend helps me feel better.)
La organización ayuda a las personas sin hogar.
(The organization helps homeless people.)
Este manual te ayudará a entender el programa.
(This manual will help you understand the software.)

Asistir

B1
Asistir has two primary meanings. First, it can mean to assist in a more formal, professional, or official capacity, often implying a secondary or subordinate role (like a nurse assisting a doctor). Its second, and far more common meaning, is to attend or to be present at an event, class, or meeting. This is a major false friend for English speakers.
La enfermera va a asistir al cirujano durante la operación.
(The nurse is going to assist the surgeon during the operation.)
Todos los ciudadanos tienen el derecho a ser asistidos por un intérprete.
(All citizens have the right to be assisted by an interpreter.)
No pude asistir a la reunión de la mañana.
(I could not attend the morning meeting.)
¿Cuánta gente asistió al concierto anoche?
(How many people attended the concert last night?)
Es obligatorio asistir a todas las clases para aprobar el curso.
(It is mandatory to attend all classes to pass the course.)

Summary

In short, use ayudar for everyday helping in almost any situation. Use asistir when you mean to assist in a formal or professional context. Most importantly, remember that the most frequent use of asistir is to attend, a meaning that ayudar never has.