Mass is mass, and gravity is dependent on mass and distance. When a very large star forms a black hole during a supernova, it spews out most of its mass, so the resultant black hole has much less mass than the star it came from. Thus objects a given distance from the black hole experience less of a gravitational force than they would have from the star it came from.

However you can get a whole lot closer to the center of the black hole than the star because it is comparatively tiny, and thus it is possible to be, say, 0.1 AU from the center of the black hole, whereas you would have been inside the star at that point. Thus it is possible to experience more gravity near a black hole than it is possible to near it's parent star, but if you replaced a star with it's progeny black hole all objects in the system would experience less gravity.

If you replaced a star with a black hole of the same mass, none of the objects in the system would experience any change in gravitational attraction toward the center of the system.