The term "nationality" has several different meanings depending on the context. I am assuming we are talking about the legal concepts of "nationality" and "citizenship".
In the legal context, "nationality" means an internationally-recognized relationship between a country and a person that, among other things, allows the person to hold the passport of that country. "Citizenship" is defined internally within each country to mean a status that (supposedly) allows the person full political rights in the country.
A citizen of a country is always a national of a country, but not the other way around. Although in many countries, all nationals are citizens, in some countries, there exist non-citizen nationals. Here are a few examples:
- People born in American Samoa are automatically US nationals at birth, but (unless they have a US citizen parent) are not US citizens at birth. As non-citizen US nationals, they hold US passports, and can live and work in the US without restriction, but they cannot vote. Other US territories were previously in a similar situation until Congress extended citizenship to them; but Congress has not extended it to American Samoa.
- There are currently 6 classes of British nationals. In addition to British citizen, there are also: British Overseas Territories citizen, British Overseas citizen, British subject, British National (Overseas), and British protected person. They all hold British passports but have different rights; some of the latter statuses do not have right of abode in the UK.
- In many Latin American countries, for example Mexico, nationality (nacionalidad) is acquired at birth, but citizenship (ciudadanía) is limited to those who have turned 18 or 21. So children would be non-citizen nationals of the country.
When talking about foreigners, generally it is only nationality that one cares about, because whether you are a citizen or a non-citizen national of a country is really not relevant to another country. All they care about is that you travel on the first country's passport, and are afforded the consular protection of that first country. However, domestically within a country, one often talks about citizenship, because some of the domestic rights and obligations of citizenship, e.g. voting rights, taxation, right of abode, etc., may be different for non-citizen nationals.
Unfortunately, due to the fact that many countries don't have non-citizen nationals, and even in the countries that have them, they are rare or the status is not well known, most people confuse the terms "nationality" and "citizenship", or exclusively use the term "citizenship" even when what they mean is "nationality".