I would not say they are the same things but, as traditionally understood by most premodern Sunni Muslim jurists, the Caliphate was (theoretically) a unitary imperial polity. So I guess you could call the Umayyad, Abbasid, Fatimid, and Ottoman Caliphates "empires."

Caliphs are by definition religious leaders, so they're kind of like Pope and emperor rolled into one. The Caliph wasn't even always the de-facto political leader and was occasionally reduced to a ceremonial role, e.g. under the Mamluks. Of course, this is true for some emperors as well. (There are also Sufi orders that call themselves caliphates without any implied political leadership, but this is getting into semantics.)

The combination of religious and political leadership is not unique to the Islamic world at all and happened in many cultures. Prior to the Italian unification the Pope was a political leader, too. Queen Elizabeth II is still the Supreme Head of the Church of England and Emperor Akihito is still the spiritual leader of Shinto.

So yes, the caliphate is basically the Islamic version of a theocratic empire. Which doesn't say much because empires come in all shapes and sizes.

In 19th century Europe, with the advent of imperialism and the waning influence of religion on politics (and the dissolution of the HRE) the Ottoman Empire for all intents and purposes operated very similarly to the modern western empires of the time. However, by then the power of the Ottoman sultans was primarily political with little focus on religious leadership and the title of Caliph. In the late 19th century, Sultan Abdul-Hamid II was briefly successful in consolidating political power by reasserting his status as Caliph, but after a military coup in 1908 the Ottoman Empire was essentially turned into a constitutional monarchy.