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.When comparing far distances, I would prefer to use nearer and nearest, not closer to and closest to.
There must be a web-site somewhere [probably posted already, before I hit the Submit button] that lists the other possible syntaxes for an English sentence (seven in all, I think).
Close (clohs) is an adjective that means near to, within a short distance, or to be emotionally connected. As a verb, to shut can usually be replaced by to close.