Both means the same which is 'to' in English. But in my personal opinion,에게 sounds like more familiar and cute while 한테 feels like just a grammar.

The honorific form for "한테" is also "께". 에게, 한테, and 께 indicate that the preceding noun is the indirect object of the action. An indirect object is someone or something that you are doing something to, such as giving a gift to your friend. 에게 and 한테 are casual, however 한테 is used more in speech while 에게 is usually written. 께 is honorific and thus is used for people of higher status than the speaker. 께 does not entail that the verbs in the sentence necessarily need to be conjugated politely. The verb conjugation depends more on the status of who the speaker is speaking to, not who the indirect object is. Lastly, 에게, 한테, and 께 are used for people or animals, not for inanimate objects. 부모님께 전화를 했습니다. I called my parents. 저에게 한국말을 가르쳐 주세요. Please teach Korean to me. 친구한테 돈을 빌려 주었어. I lent money to my friend. 할머니께 물어보세요. Please ask Grandma.