As in the topic, I would like to know what are the differences between 游学 and 留学. As far as I know a student going studying abroad could use any of them. Could he? When aren't they interchangeable?
to study abroad
'留' = 'stay'; '學' = 'study'
'留学' means 'stay (abroad) to study'
If you say: "我要去留學" (I am going to study abroad), it implies there is a clear destination. e.g. "我要去美国留学" (I am going to America to study). In this case, you are going to enroll to a particular college or university in a particular city in a foreign country
'游学' is short for 一邊游历一邊學習 (learning while traveling). It is a very general term.
'游学' could mean you are traveling across your own country and learn new things as you go, from serious study like taking college courses, to casual learning like learn to make wonton noodle soup from a restaurant owner; It could also mean you go to many foreign countries and studying in many different universities or taking apprenticeship under different masters.
'四方游学' is a common phrase that describe 'go to many places to study' or 'traveling the world and study along the way' . But there's no such thing as '四方留学'
Nowadays, both 遊學 and 留學 mean "study abroad" but their periods are different.
遊學 means "travel and study". It is usually brief and only lasts from a few days to a month. It is usually arranged by some travel agency or something alike under the name of 遊學團. The content varies.
留學 means "stay and study". It usually spans over from one year to several years and stay in school, college or university.
Note: Simplified Chinese merges two characters 遊 (travel) and 游 (swim) to 游. And that means 遊學 is simplified to 游學 in mainland China.