They are the same kind of storm — a tropical cyclone — and the name depends only on where it forms. In the Atlantic and the eastern Pacific it is a hurricane. In the western Pacific it is a typhoon. In the Indian Ocean and around Australia the same system is simply called a cyclone.
The core difference
There is no meteorological difference. A rotating tropical storm with sustained winds above roughly 74 mph gets a different name in each ocean basin, and that is the whole distinction.
- hurricane — North Atlantic, Caribbean, Gulf of Mexico, eastern Pacific
- typhoon — western Pacific, affecting East and Southeast Asia
- cyclone — Indian Ocean and the South Pacific
Where the words come from
Hurricane came into English through Spanish from a Caribbean Taíno word, hurakán. Typhoon has a tangled history, with influence from Chinese tai fung, great wind, and from Greek typhon, whirlwind. The words arrived in English by different routes because English speakers met the storms in different oceans.
The technical term
Meteorologists use tropical cyclone as the umbrella term, which is why forecasters can describe a storm crossing from one basin to another without contradicting themselves — the system does not change, only the label.