In most cases it does not matter if you use American or British English, the most important thing is to be consistent across a document. I had largely advocated using British English in Europe, but now with Brexit …
When writing a scientific paper, check to see if the journal includes a preference in their guidance for authors.
There are some excellent online resources discussing the differences …
Oxford Dictionary [accessed 4.8.2017]
Available from https://en.oxforddictionaries.com/usage/british-and-american-terms
This includes an extensive list of words and another of phrases.
Oxford Dictionary also has a good article (and video) about British vs American spelling: https://en.oxforddictionaries.com/spelling/british-and-spelling According to the Oxford Dictionary: “The differences often come about because British English has tended to keep the spelling of words it has absorbed from other languages (e.g. French), while American English has adapted the spelling to reflect the way that the words actually sound when they're spoken.”
Babbel Magazine has an excellent article entitled, “What Are The Differences Between American And British English?” [accessed 4.8.2017]
Available from https://www.babbel.com/en/magazine/what-are-the-differences-between-american-and-british-english
It includes a comprehensive list of the kinds of differences in spelling, vocabulary, and grammar.
“Six Differences Between British and American English” [accessed 4.8.2017]
Available from https://learningenglish.voanews.com/a/six-difference-between-britsh-and-american-english/3063743.html
This very good online article provides an explanation for why American spelling is different:
“There are hundreds of minor spelling differences between British and American English. You can thank American lexicographer Noah Webster for this. You might recognize Webster’s name from the dictionary that carries his name. Noah Webster, an author, politician, and teacher, started an effort to reform English spelling in the late 1700s. He was frustrated by the inconsistencies in English spelling. Webster wanted to spell words the way they sounded. Spelling reform was also a way for America to show its independence from England.”
Another very good article: “Differences in American and British English grammar – article” [accessed 4.8.2017]
Available from http://www.onestopenglish.com/grammar/grammar-reference/american-english-vs-british-english/differences-in-american-and-british-english-grammar-article/152820.article
An interesting analysis of “Gray or Grey” [accessed 4.8.2017]
Available from http://blog.dictionary.com/gray-or-grey/
There are other Englishes as well! The BBC posted an article on “Why Canadian English is Unique” [accessed 4.8.2017]
Available from http://www.bbc.com/culture/story/20150820-why-is-canadian-english-unique
Even Nature enters the fray with this article entitled: “Text mining uncovers British reserve and US emotion” [accessed 4.8.2017]
Available from http://www.nature.com/news/text-mining-uncovers-british-reserve-and-us-emotion-1.12642