What is the difference between the adjectives close and near? Are they totally synonymous? Is there some nuance that I'm missing?
As a native speaker of Spanish, I can't see any difference, since both are translated to cercano.
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
They are mostly interchangeable, but there is some difference. See this Usage Note taken from the OALD:
The adjectives near and close are often the same in meaning, but in some phrases only one of them may be used:
The near future ◇ a near neighbour ◇ a near miss ◇ a close contest ◇ a close encounter ◇ a close call.
Close is more often used to describe a relationship between people: a close friend ◇ close family ◇ close links. You do not usually use near in this way.
Close and near have similar meanings. There are times where you can use close or near, but in some phrases you don't use close instead of near, or vice versa.
The conflict is unlikely to be resolved in the near future.
An all-electric future was near at hand.
She needs to keep a close eye on this project.
She came close to calling the President a liar.
Pay close attention to what your body is telling you about yourself.
They are pretty much synonyms, but you can't use them the exact same way everywhere. Depending on the sentence, there can be different connotations. For example
Anna and Bob are close to each other
Anna and Bob are near each other
are not the same. The former can imply that they are located close to each other, but it can also imply a close relationship, they are good friends (not quite lovers though, just good friends).
The latter only implies close distance in space.
In general, my feeling is that 'near' is more of a technical term, usually implying a short distance in physical space, whereas 'close' can, in addition to short physical distance, also refer to all kinds of distances, such as emotional relationship (see example above), temporal near-coincidence ("a close call" for two events that happened nearly at the same time), a difference in meaning ("the explanation is close enough to the truth"), etc.