In a linguistic perspective both words mean the same however in Hindu context they are dramatically opposite. The modern usage of the word "History" happens in the context of Euro-centrism. The subject of history as it is taught in schools and universities is influence by European education model. Europ's history had several components. One important component was inheritance and class. It mattered who your father was and how you are linked to the rich and powerful. Nations faught and signed treaties. They developed their own system of law and contracts.
All this made "facts" of supreme importance. It mattered what paper was signed on what day with whom as a witness. The exensive record keeping gave rise to modern version of civil servants but colonialism also made it extremely critical for Europe to ensure records are kept of every even aroung the world. But over time people realized thay they could look at the records and reconstruct the events of past. This became the academic stream of "History". As the colonizers reached India and Africa, they used the similar methods to reconstruct India's history. Europeans also realized that even nature keeps records. Soil, trees and fossiles store information that can tell us about our past. This field became archeaology. Even today historians of western world deeply obsess over these factoids.
The dharmic system was entirely different. In Hindu religion we do not care much about records of the past but we rather focus on the substance that needs to be tranfered to future generations and we call that "Itihasa".
Itihasa is not supposed to be a central repository of extensive record keeping for all factoidsm, rather it is a system if keeping only those records that are important for future generations in a form that are worth transfering. Do not be mistaken, Hindus cared as much about contracts, ownerships, geneology, rules etc. Hindu Puranas might look likes stories to us but they are also full of dates, records, birth records, death records, geographical information, laws, rules, customs, geneology and so on. But they are also full of poetry, imagination and fantasical details.
The story of Ramayana is a "itihasa" not because Hindus care about the very exact location where mother Kaushalya gave birth to Ram. In christian religion or Islam it is absolutely crtical to find out where Jesus was burried or where the Kaba is. For Hindus it is just a matter of belief. We believe Rama was born in Ayodhya but there is no need to prove it as a fact because, over thousands of years we have known it to be Ayodhya.
It does not matter if Hanuman was an actual monkey or whether he jumped to Sri Lanka. It is however important to know that Hindus knew the geography of entire Indian contient well. There are Vidhyas, Wester Ghats, Rameshwaram, Kashi, Sri Lanka, Jharkhand, Chattisgarh etc. mentioned in Ramayana with different names. Ramayana passed that knowledge from generation to generation through the stories. Every kid knew it.
Had our forefathers kept this as some kind of boring record keeping mechanism were King X waged war on King Y over a woman, this would have been boring and quickly forgotten. Also, what is the point of remembering all this ? What is the point of elavating Rama to the status of a divine avatara ?
We automatically see that answer. Ramayana as a Itihasa has served the purpose no boring history book could have served. Not only every kid in India knows about Rama's victory on Ravana they actively seek to emulate Ram. Sometimes it is the victory of your inner good over your inner evil but then it is also an inspiration for an entire generation to rebel again tyranny.
Islamic invaders and the British both failed in India for this very reason. Our Itihasa is not in soem boring book that can be burned. It is not kept by some pompus clerks who can be killed off. It is evolved with us and with all of us and hence continues to live.
Itihasa thus has served the purpose it was meant to serve far better than the modern "History".