What's the difference between a state and a country? Is there any precise political definition that, perhaps, the UN would follow?
What's the difference between a state and a country?
There's no hard and fast rule but independent states are the units that are most usually parties to international treaties, conduct foreign relations and so on.
For most places, the concepts of state and country apply to the same territory but there are very many exceptions.
There is a state called Georgia which is not an independent state, it is a constituent state of the United States of America. There is also a place called Georgia which is an independent state.
The UK is a state but England, Scotland, Wales and Northern Ireland are usually considered countries. They are not (currently) independent states although at least England and Scotland were in the past. Wales came pretty close.
Sometimes Wales is described as a principality and sometimes Northern Ireland is described as a province.
Is there any precise political definition that, perhaps, the UN would follow?
The UN, and other bodies, have to be pretty flexible (i.e. vague) in order to accommodate all sorts of complicated political situations. For example consider the Republic of China (ROC) and the People's Republic of China (PRC); The Principality of Monaco - "a sovereign city-state, country and microstate"; Vatican city; The European Union; etc etc.
The UN occasionally play fast and loose with terminology too, consider their use of "country" and "state" in the following
How does a country become a Member of the United Nations?
Membership in the Organization, in accordance with the Charter of the United Nations, “is open to all peace-loving States that accept the obligations contained in the United Nations Charter and, in the judgment of the Organization, are able to carry out these obligations”. States are admitted to membership in the United Nations by decision of the General Assembly upon the recommendation of the Security Council.
Most often though, UN members are referred to as states.
Note that the EU is an "enhanced observer state" at the UN. EU member states are also individually UN member states. There are many complications to UN membership.
In short: no. Although there is a concept of "state" in international law, it is somewhat vague and hard to pin down. There is no legal definition of "country" at all, but sometimes the United Nations uses that word to indicate all of the things that happen within a state's boundaries.
For example, Canada is a state under international law. When discussing the economy of Canada, you might use the word "Country" because in strict terms a state does not have an economy. The economy is not a part of a state, but it operates within the boundaries of the state.
What is a state in international law?
This has been discussed in many answers. For example, see my answer here. The Montevideo Convention defines a state as:
The state as a person of international law should possess the following qualifications:
a ) a permanent population;
b ) a defined territory;
c ) government; and
d) capacity to enter into relations with the other states.
Strictly speaking, this definition doesn't apply to anything other than the Montevideo Convention. However, it is often invoked in discussions regarding statehood, even outside of this Convention. It's as good as it is going to get.
What is a country?
There is no definition of a country in international law. However, you will see the term appear in UN reports such as this one. There is nothing legalistic or technical about this use of the term. As an informal term, it appears to be used when referring to activities within a state's boundaries, but which are not properties of the state.
For example, the linked report is discussing economies. Most states do not have an economy themselves, the economy is something "outside" the state.