The three terms sister city, twin town and friendship city are equivalent. (The phrase twin city is usually reserved for two cities that are so close as to be virtually indistinguishable except for being historically located in different governmental regions.)
Different places use different words for a relationship between two cities that involves formal links between the local governments for their mutual benefit. The links are often made because of some common cultural heritage, such as a similar name or geographical proximity, or some other jointly-perceived reason to feel that there will be benefits, either economic or on some other level.
https://www.casey.vic.gov.au/council/our-city/sistercities/sister-citiesTwin towns or sister cities are a form of legal or social agreement between towns, cities, counties, oblasts, prefectures, provinces, regions, states, and even countries in geographically and politically distinct areas to promote cultural and commercial ties.
https://en.wikipedia.org/wiki/Twin_towns_and_sister_cities
Some cities have multiple sister cities. Melbourne (Victoria, Australia), for example, is sister with Osaka, Japan; Tianjin, China; Thessaloniki, Greece; St. Petersburg, Russia; Milan, Italy and Boston, Massachusetts, USA. Some of its smaller divisions (such as the council of Casey linked above) have their own sister city relationships, both within Australia and internationally.
http://www.onlymelbourne.com.au/melbourne-sister-cities#.WIrpfFze2ug
Sister Cities International is a group that coordinates official sister city affiliations. According to a pdf document that you can find on their site, but to which I cannot provide a direct link, when either or both of the cities decides to an discontinue active relationship, the link is renamed as emeritus, indicating that it was once an active partnership, but is not now.
https://sistercities.org/about-us/what-is-a-sister-city-3/
