What is the difference between a country and a principality?

Principality - A localized territory or region ruled by a prince (or princess), from which the term is derived. These may be countries. A prince may rule more than one principality. All principalities can be grouped under the general category of 'state.' A principality is ruled autocratically and is therefore distinguished from a republic, the only other type of state

Ecclesiastical principalities - A principality technically under the rulership of a prince, but nonetheless strongly dominated by the Church.

Hereditary principality - A principality ruled by a prince whose family has controlled the principality for several generations. Hereditary principalities, according to Machiavelli, are generally easy to rule and maintain.
http://www.sparknotes.com/philosophy/prince/characters.html

Andorra, Liechtenstein and Monaco are principalities.

Wales

{Liechtenstein;} Monaco.

Asturias and Andorra.

A principality is a country which is governed by a prince, e.g. Monaco or Liechtenstein.

A country is any area which is a distinct geopolitical entity: it may be a principality, a kingdom, a republic or whatever. Some reference books limit the use of 'country' to sovereign states, and call colonies and dependencies 'territories' or something like that.

Generally recognised surviving sovereign principalities are Liechtenstein, Monaco, and the co-principality of Andorra. Extant royal primogenitures styled as principalities include Asturias (Spain), and Wales (UK)
https://en.wikipedia.org/wiki/Principality

A principality is a country which is ruled over by a Prince.
https://en.wikipedia.org/wiki/Principality