The subtle difference lies. I'll try to explain 'have been' and 'went' irrespective of the answer stated there that might have confused you.
Have been is generally used to say you had gone to that place and come back. On the other hand, went in this context talks about your presence on that place in the past.
I have been to America
...means at least once I had gone to the states and come back.
Think about this - if your friend asks you, "Have you gone to America?" This would be unnatural to many. On the other hand, if he wants to ask whether, in your lifetime, you have ever visited America, the better construction is "Have you been to America (ever)?"
So, to answer this in the context of visiting the places, if you want to say that you had visited that place and had come back, the better way is have been and if you are more concerned about your presence in past, use went as in " When I went to New York City, I learned that Americans there are very amicable".