Closest between point and linestring is the point itself, but closest point between a linestring and point is the point on line string that is closest.

SELECT ST_AsText(ST_ClosestPoint(pt,line)) AS cp_pt_line,
	ST_AsText(ST_ClosestPoint(line,pt)) As cp_line_pt
FROM (SELECT 'POINT(100 100)'::geometry As pt,
		'LINESTRING (20 80, 98 190, 110 180, 50 75 )'::geometry As line
	) As foo;


   cp_pt_line   |                cp_line_pt
----------------+------------------------------------------
 POINT(100 100) | POINT(73.0769230769231 115.384615384615)
				

closest point on polygon A to polygon B

SELECT ST_AsText(
		ST_ClosestPoint(
			ST_GeomFromText('POLYGON((175 150, 20 40, 50 60, 125 100, 175 150))'),
			ST_Buffer(ST_GeomFromText('POINT(110 170)'), 20)
			)
		) As ptwkt;

                  ptwkt
------------------------------------------
 POINT(140.752120669087 125.695053378061)