A school trip could be to any location for any reason, either recreational or educational. For example, if teachers take a group of five-year-olds to a theme park or circus, this would count as 'school trip'. A 'field trip' is educational. It's an opportunity to study in the 'field', as opposed to in the classroom. A typical field trip might be to study a topographical feature for geography, or to visit an archaeological site for history.

"Field trip" can also be used in colloquial conversation for a humorous effect in the sense of describing something or going somewhere as a short adventure; trying to make a mundane trek sound like it will be fun and exciting.

In the UK you could use both terms for a school trip or outing. 'School outing" may be an older term you will hear some people say. A "field trip" would be used to describe something like a scientific project by scientists and researchers. A field trip when used by a school or education facility would describe, going somewhere and/or doing something for educational purposes.

As an American they would both mean the same thing