Was Jizya tax an equivalent of Zakkat for non-Muslims?

No no, not so much. Jizya's goal was never really to replace zakat per se. Zakat is charitable giving to the poor, a 2.5% tithe on a Muslim's yearly income which is intended for the less fortunate in society. A wealthier Muslim can also create a waqf (pious endowment); this could be something like building a mosque, or establish a business to which all of the profits will go towards a local mosque, or food for the poor, etc. Zakat is also one of the five pillars of Islam, making it a hugely important part of the religion and one that is not really up for debate, unless you are below an certain income threshold, in which case you dont pay at all.

Jizya on the other hand has a more complicated history and opinions on it have varied throughout history. The verse in the Quran which mentions it (Quran 9:29) does not actually specify what jizya is, it only mentions that it is something that people of the book (Christians and Jews) living in Muslim lands must do.

"Fight those who do not believe in Allah or in the Last Day and who do not consider unlawful what Allah and His Messenger have made unlawful and who do not adopt the religion of truth from those who were given the Scripture until they give the jizyah willingly while they are humbled."

We thus must look at the Hadith as well to get an actual definition of what jizya is. We have Hadith regarding Muhammad and the early Rashidun Caliphs collecting jizya payments from people of the book living in Muslim lands, and there is a history of Islamic legal commentary deciding what is and is not acceptable jizya practice based upon the four legal schools.

We also have to look at what jizya's historical use was. If jizya was merely a form of zakat, then it would, like zakat, be designated as for the poor. However, jizya was always, even in Muhammad and the Rashidun Caliphate's times, given to the "state", such as it existed at the time. As the Islamic empire expanded, different jizya "deals" were constantly being made with different groups as they were conquered; as such there was never any uniform jizya payment for Dhimmi (Non-Muslim) populations across the Islamic world (however, it is important to not that, unlike zakat, jizya must always be paid, poverty still requires a Dhmmi to pay jizya). Some Islamic empires would rely heavily on jizya payments, and some (such as the Ummayads) would even discourage conversion to keep populations paying an increasingly crucial tax. The Ottomans would apply jizya differently in different regions, and the Eastern European Christian slave boys that were taken to become the elite Janissaries were a form of jizya, organized through the devshirme system.

So to sum up, I would have to say that the fundamental difference between jizya and zakat is its intention: zakat is a more personal tithe for the Muslim faithful to be given to the poor; jizya is a tax, not even necessarily monetary, levied upon non-Muslim populations to be given to the state to further the aims of the state. Zakat is an expression of religious piety, whereas jizya is more of an expression of submission to the state. While few Muslims would argue for jizya today, save for more fundamentalist/political Islam/certain salafist types, it is very important that we keep in mind that it's a quite progressive system compared to how other conquered religious groups have been dealt with in similar societies. While its obviously a bit irritating for me, poor Christian peasant living in a Muslim empire, to pay my jizya tax to the Abassids, Fatimids, Safavids, Ottomans, whoever, Id be much happier to pay jizya than, say, a violent forced conversion or extermination as in much of Europe at the time!

Sources: Hourani. A History of the Arab Peoples

P. Bearman, Th. Bianquis, C.E. Bosworth, E. van Donzel and W.P. Heinrichs. The Encyclopedia of Islam

Kennedy, Hugh. The Prophet and the Age of the Caliphates, 2nd Edition