근처 implies a bigger area, that something is in the area. 가까이 more so implies that fact that something is very close or near.
when i asked my friend she just told me 가까이 is "near" and 근처 is more like "around", closer in meaning to 주위
I know English articles are difficult, but in most sentences, both "a" and "the" should be acceptable. Not just either/or.