Learn English – “nibble” vs. “nibble on”

Close and near can be used as adjectives, and in many cases they are the same when talking about physical distances.

The train station is close.
The train station is near.

They are not the same when talking about more abstract concepts, like relationships.

My mother is close [she is 3 feet away].
My mother and I are very close [we have a strong family bond].
My mother is near [she is 3 feet away].
My mother and I are very near [we are 3 feet apart].

Only near can be used as a preposition without the word to.

*We are close the train station.
We are close to the train station [correct: close to is the proposition].
We are near the train station [correct: near is the preposition].

When you convert to adverb form, they are not interchangeable at all. In this case, closely implies "at a small distance", while nearly implies "almost but not quite"

*We are nearly following the news.
We are closely following the news.

*I closely hurt myself.
I nearly hurt myself.

* incorrect usage

I am surprised that your girlfriend finds "swim teacher" more natural: In my (also UK) idiolect it is impossible.

"X teacher" (meaning "teacher of X"), X is usually (I would have said always) a noun, and nearly always an uncountable abstract: swimming, physics, calligraphy, elocution, French, carpentry, driving. The cases I can think of where it is countable are musical instruments (violin, piano), but in these cases you can paraphrase as "teacher of the piano", where "the piano" is a sort of universal, and hence again uncountable even though it "piano" is normally countable.

"Swim" as a noun is countable (even if the plural is rather rare), meaning "single event of swimming", and so does not fit into the pattern for me.