The terms ministry and office can sometimes be used synonymously in the church, but there is a distinction between the two, and mistaking them as being the same can lead to misunderstanding.
Ministries are positions of service to the Church, for the Church, by the Church. They are positions of service that come from God’s provision of gifts and calling, whether officially recognized by church leadership or not.
Offices are official designated positions in the Church. They are official appointments that are given based on leadership authorization and congregational acceptance.
God is (or at least should be) consulted in prayer and followed in execution of both, but a minister works out their calling whether church leadership approves of it or not, while an office can only be obtained with the approval of ones peers.
As an example, parts of the western church do not recognize the prophet as an office, and some also deny it as a ministry, but the prophet’s ministry responsibility to bring God’s word of correction to the Church remains nonetheless. As in history, the leaders may not like what the prophet has to say, and may even rebuke, condemn, or otherwise denounce the prophet. Just because the church leaders don’t like what the prophet has to say does not excuse the prophet from carrying out their ministry as directed by God.
On the other hand an office may be handed out to anyone who technically meets the qualifications. And an office can be removed from a person who falls out of those qualifications. The example that comes to mind is from history, when fathers pushed their sons to go to college and become ministers in the church because it was a high paying and prestegious position - whether the son actually felt called to ministry or not. Similar to how parents will push their kids to become doctors or lawyers today. And, similar to how politics still works today, if the father could bribe someone in the church then he could effectively buy his son’s way into those prestegious postions in the church.
Both offices and ministries are ordained in the Bible, and have their purpose in God’s plan. And the words can sometimes be used interchangably depending how they are being spoken of - such as a Pastor (Shepherd) also being called an Elder. The twist on each comes from our sin-filled struggle to execute either one faithfully.