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Roughly speaking, 周り is 辺り + an imaginary circle. 周り means the area that belongs to 辺り and that is outside or along the circle.
辺り means "vicinity" or "neighborhood" in general while 周り is "vicinity" or "neighborhood" with the connotation of "circumference," "girth" or "surrounding area."
Maybe some examples may help a bit:
When you're talking about beaches, seasides or port cities in general, you can use 海の辺り. If you're referring to coastal areas and picturing the coast line in a map in your mind, you say 海の周り; you're talking about the area along the coast line.
When you say この海の辺り while, say, talking with your friends on a deck of a ship in the middle of the Pacific Ocean, it can mean "this area (of sea)." When you use この海の周り in the same situation, it can mean "neighboring areas."
この辺りは危険 means "this area is dangerous." If you say この周りは危険, the place you're in might be safe, but neighboring areas are all dangerous.