What is the difference between "Country of Issue" and "Country of Citizenship" of a passport?

Rather unusual example, but you could be travelling on either Holy See or the Sovereign Military Order of Malta passports, while you are (obviously) a citizen of your home country. These entities do not have the concept of citizenship in the usual sense, indeed, they are not countries.

Any country can emit a passport for any person regardless of its citizenship.
Case in point, France does not recognize a passport as a proof of citizenship for its own citizen. (In BOLD here and here for french readers)

The passport is an official travel document stating to other countries that the holder is recognized as specified. Having a passport from a certain country entails some prerogatives, like being exempt of visa, or having a passport automatically purported to be valid 6 month after expiration etc….

But holding a passport and having a citizenship are in theory completely different notions. Although practically, 99.9% of the passport are issued by the country of citizenship, there are no international laws to my knowledge which prevent a country to issue a passport to non-citizen, (although some local legislation may prevent doing so.)

exemple : UK passport for Hong Kong, US passports, any diplomatic passport, which are issued on a mission basis for representation of the said State, who is sovereign in deciding who represents it….

It’s difficult to know precisely without context but there are few cases where this could be relevant, in particular:

• Travel documents other than passports (e.g. refugee travel documents) are issued by the country where the person resides and indicate that the holder can be readmitted to the country of issue even though they are still citizens of another country. Formally, these documents are typically not called “passports” but they look like one and are often informally called “refugee passports”.
• Some countries like the UK and the US have a distinction between “citizens” and “nationals”. For the US, some people from the American Samoa can for example get a passport issued by the US Department of State but cannot claim the US as their “country of citizenship”.
• People living in countries that are not recognized as such by many other countries in the world (e.g. Palestine, Northern Cyprus) sometimes travel with a passport from another country (e.g. Jordan, Turkey) where they might or might not be considered a citizen. However, it would be difficult to accept “Palestine” as “country of citizenship” if it’s not officially recognized wherever you are so that’s probably not the intent of the form.

On the other hand, if you have a regular passport, issued by your country’s authorities, then I would tend to think that “country of issue” and “country of citizenship” are the same even if you got your passport through a consulate abroad (that’s certainly the terminology used by the US authorities, as shown in the link provided by Mark).

Although not common, some countries issue passports to non-citizens as well. As you may have noticed, the data page of a passport often states the nationality or citizenship of the holder in a separate field and the citizenship may actually differ from the issuing country.

One example is laissez-passer documents or emergency passports, which may be issued by foreign governments. For example, if you travel to a country without a diplomatic representation from your home country and lose your passport, your home country may have an agreement with a third country to help you with a temporary emergency passport.

Another example is the British Passport, which can be issued to all British nationals, even if they are not British citizens (British nationality law makes a clear difference between “nationals” and “citizens”).

From the ESTA website (which I was reading just as you posted the question!):

Your “Country of Issue” is the same as your “Country of Citizenship”.
For instance, if you are a citizen of the United Kingdom, but are
getting your passport from the UK Consulate in Hong Kong, the UK is
your country of issue. The UK Consulate may be located in Hong Kong,
but Hong Kong is not the country issuing you the passport.

However, it’s worth noting that some forms word this differently, and ask for place of issue or authority who issued it. Then it could be different.

For example, my current New Zealand passport.

Country of citizenship: New Zealand, clearly.

But where it was issued? I was in London, UK, and the field in the passport says “DIA LON”. (Department of Internal Affairs, London). This is the Identity Services Office in London – often referred to as the London Passport Office – and is responsible for the issuing and renewals of New Zealand Passports in the United Kingdom and Ireland.

So when it says place of issue, that’s usually what gets put down for me, or when it asks for ‘Authority’ as that’s the specified field in the passport as well.

Country of Issue, however, would still be New Zealand.