In England now, 'academy' has a particular meaning in the secondary school system, Academy (English school) - Wikipedia, the free encyclopedia.
However, apart from this, there is no officially recognised difference between: 'school', 'college', 'institute', 'academy'.
ps. I have just looked through lists of independent schools in Britain. Most are schools, some of these being grammar schools or high schools; some are colleges; a few, especially in Scotland, are academies. I didn't find any institutes, but there are some in the USA.