What is the difference between "vote" and "ballot"?

"A ballot" is usually more of the specific piece of paper you vote on or the individual votes that are cast. "A vote" or "your vote" is usually more conceptual or talking about all the votes in bulk, or more like the whole election itself. Basically: to have your vote counted you cast a ballot. But language is fuzzy and both words can be used for the other situation. And "ballot" the noun can be verbed. Like saying "we held a secret ballot" doesn't sound wrong or off to me. I'd use "we held a secret vote" ahead of "ballot" for example.

In current usage, “secret ballot” is a technical term, so it is not replaceable with “secret vote”, regardless of which one makes more sense. That said, there is a logic here. Ballot is always a noun, white vote can be a noun or verb. Sounds like you know this already. But it’s also the important difference here: a secret *vote* sounds like the act of voting is itself a secret; IOW, that there is a vote at all is a secret, or that the underlying data is not auditable, or who is voting is secret. It has a furtive or malicious implication. A secret *ballot* OTOH means that the physical ballot is a secret, which is the point here: everyone can watch you go inside and vote, but no one can see who or what you voted for.

"Vote" is a verb in most uses (though it can be used as a noun on occasion). "Ballot" is a noun in most uses, though sometimes it may be used as a verb. Also, using the same noun, name, or verb twice in one sentence is usually very awkward in English with very few exceptions. Using both "ballot" and "vote" here avoids a lot of the awkwardness, the only question the author had to consider was which to make the noun and which to make the verb.

They *can* be synonymous, in some contexts. Usually, a ballot is writing your choice on paper, putting it in a box, to be counted later. A vote can be a quick show of raised hands. A group of friends casually choosing a restaurant may "take a vote on it". They wouldn't organize a ballot.

There is much overlap in usage. But a vote includes: the official choice made by a person in an election; and the physical way a person makes that choice whether it's by show of hands, orally, by computer, marble, or paper. To have or hold a vote means to decide something by voting. A ballot is the physical thing, usually paper, used to cast a vote. Ballot can also mean then choices available to a voter in an election. Balloting or to hold a ballot can mean to hold an election. In your example, the physical ballots, and therefore the choice of any particular voter is secret. But the fact of the election and the outcome is not. A secret vote suggests a clandestine election, not merel privacy for individual votes.