Learn Russian: Врач запретил ему пить вино, зато разрешил пить сок. - The doctor forbade him to drink wine, but on the other hand allowed him to drink juice.

The verbs запретить (to forbid) and разрешить (to allow) both require the Dative case for the person receiving the command or permission. Since they belong to the group of verbs of influence and assistance, we use the Dative pronoun ему (to him), rather than the Accusative его.

Why do we use the imperfective verb пить when the main verbs запретил and разрешил are perfective?

The doctor is forbidding and allowing a general habit, not a one-time event. When we talk about doing or not doing an action in a broad, habitual sense, we use the imperfective infinitive like пить, regardless of whether the main verb that controls it is perfective or imperfective.

Could I just use но instead of зато here?

You could use но, but зато adds a specific nuance of compensation. It translates closer to "but on the other hand" or "but to make up for it." It shows that being allowed to drink juice is a positive thing that balances out the bad news of not being allowed to drink wine.

Do I need to repeat ему before the second пить?

No, you do not. Because ему was already established in the first clause and the person being spoken about has not changed, Russian naturally drops the pronoun in the second half of the sentence to avoid sounding repetitive.