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
and
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.