What's the grammatical difference between a transitive and intransitive verb.?

Grammatically, there's no difference. They're both verbs. An intransitive verb does not take a direct object. There's no word in the sentence that tells who or what received the action. A transitive verb takes a direct object.

A transitive verb cannot have a direct objective. He was hitting her. Her is a direct object and so hitting is transitive. He went home. Went cannot have a direct object and so is intransitive.

A transitive verb is one that is used with an object: a noun, phrase, or pronoun that refers to the person or thing that is affected by the action of the verb. In the following sentences, admire, maintain, face, and love are transitive verbs:

I admire your courage.

We need to maintain product quality.

I couldn’t face him today.

She loves animals. ........more.....

https://en.oxforddictionaries.com/grammar/transitive-and-intransitive-verbs

A transitive verb is a verb that requires one or more objects. This contrasts with intransitive verbs, which do not have objects. Transitivity is traditionally thought as a global property of a clause, by which activity is transferred from an agent to a patient.

https://en.wikipedia.org/wiki/Transitive_verb

https://en.wikipedia.org/wiki/Intransitive_verb

transitive verb - A verb that is accompanied (either clearly or implicitly) by a direct object in the active voice. It links the action taken by the subject with the object upon which that action is taken. Consequently, transitive verbs can also be used in the passive voice when the direct object of the equivalent active-voice sentence becomes the subject.

https://en.wiktionary.org/wiki/transitive_verb#English

intransitive verb - An action verb not taking a direct object. In English "sleep" is an intransitive verb.

https://en.wiktionary.org/wiki/intransitive_verb#English