The difference is explained in the course notes (https://forum.duolingo.com/comment/17638579). For example, see the notes for 'Past Mynd 1' - https://www.duolingo.com/skill/cy/Past-Mynd1/tips-and-notes
These "I've gone" sentences are doing my head in. In English it sounds very weird to use the present perfect "I've gone". The only situation where I can imagine it working is if you're leaving a note to someone to explain your absence. You'd never say it, because to say that someone has gone somewhere means they aren't around! You can obviously say "I've been", but that's different, meaning and I went and then I came back ...
Well it is the correct translation. There are different ways to express the concept 'I went' etc. 'I've been is quite idiomatic English that does not match up with the Welsh very well. You will need the present perfect sometimes and it is much easier than other structures because of its similarity to the present continuous.
The sentences here are a bit arbitrary, and I accept they are more natural examples, especially with places you visit, such as a shop, rather than things you attend. I would certainly say that I have gone to church, gone to the supermarket, gone to the park etc. In addition to that, there will be big dialectal differences in English. Where they are the course writers can choose to use the form closest to the Welsh to help understand the Welsh. They do, to some extent, and should do more (in my opinion) and do more in other courses such as Scottish Gaelic.