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I've just finished a writing task for an ielts practice test asking,

There are many different types of music in the world today. Why do we need
music? Is the traditional music of a country more important than the
International music that is heard everywhere nowadays?

and I'm curious if I should use "of" instead of "for".

It's not proofreading here, but I'm not sure if the rest of the statement is required for the question.

“Music is life.” I believe that we all are used to hearing this phrase from different medias in our daily lives, such as television shows, movies and radio stations. You can try imagining that if there is no background music or theme songs in movies, those movies would be less interesting and less exciting, especially horror movies. Music also affect many things including our daily activities the same way it does to movies. Although the statement is a controversial one, I have to agree that music is parts of our lives.

First, we have music everywhere and every moment, music has been developed by human all the time, and that is why we have many types of music, for example, music which were developed in the past are called traditional music and classical music. The traditional music is a representative of local people perspectives and lifestyles that we can notice from their rhythms, lyrics and instruments.

Furthermore, we also have a type of music that most people love to listen to. It’s called international music, which is also appeared everywhere in our daily lives and represents the same things that other types of music do. Therefore, it is no exaggeration to say that traditional and international music have the same degree of important.

In conclusion, despite the increasing number of people who listen to international music, the truth is actually that traditional music is also a type of recreational way for people and reflexes many aspects of how people live, only in specific areas. This means both traditional and international music are equally important; however, they are just representatives of different groups of people.

What are the differences between "representative for" and "representative of" please??

As you correctly note this is not a proofreading service, and there are many areas where a tutor could give you advice on improvements. Overall your grasp of English is good and I will confine my answers to the what and who context in your two relevant sentences.

"The traditional music is a representative of local people perspectives and lifestyles that we can notice from their rhythms, lyrics and instruments."

I would remove two preceding words to clarify what is being represented :-

Traditional music is representative of local peoples
perspectives and lifestyles that we can notice from their rhythms,
lyrics and instruments.

Also Jason is correct the first usage could sound better with represents

Traditional music represents local peoples ...

For your second usage

"This means both traditional and international music are equally important; however, they are just representatives of different groups of people."

I would use for to clarify who is being represented :-

This means both traditional and international music are equally
important; however, they are just representatives for different groups
of people.

As above I would further simplify to

This means both traditional and international music are equally
important; however, they just represent different groups
of people