Learn Hardcore Russian: Учитель задал студентам вопрос о книге. - The teacher asked the students a question about the book.

задаёт is the imperfective present (“he is asking/poses habitually or right now”).
задавал is the imperfective past (“he was asking/posed repeatedly or over a period of time”).

What is the difference between задать вопрос and спросить?

Both can mean “to ask a question,” but they focus on different aspects:

задать вопрос (кому?) literally “to set/assign a question to someone.” It’s often used in formal or educational contexts (teachers, exams).
спросить (кого? о чём?) literally “to inquire of someone about something.” It’s more general and colloquial: you спросить друга о дороге (“ask a friend about the way”).

Why is студентам in the dative case while вопрос is in the accusative?

Russian marks the recipient of an action with the dative case and the thing being acted upon with the accusative:

студентам (dative plural) is the indirect object (“to the students”).
вопрос (accusative singular, identical to nominative here) is the direct object (“a question”).
So Учитель (nominative) is the subject, задал is the verb, студентам is “to whom,” and вопрос is “what.”