Snapping polygon layer to another layer automatically

There are at least two possible solutions:

1. Use Geoprocessing / Difference

A simple approach is to create a buffer around layer B - set the buffer distance high enough that all white spaces between the two layers are covered. See screenshot: Layer A: orange; Layer B: blue; buffered Layer B: red outline.

Now use Menu Vector / Geoprocessing Tools / Difference and set the buffer layer as input and Layer A as Overlay layer. You get a perfectly fitting version of layer B, with layer A unchanged:

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Snap vertices of layer B to nearest vertex of layer A

For QGIS 3.16 or higher, there is another option. You can use basically the same procedure as described here:

The idea is to create two points layer, representing the vertices of the two polygon layers using Menu Vector / Geometry Tools / Extract vertices. Than "snap" the points of layer B to the closest point of layer A using the new overlay_nearest expression, see visual changelog. Create a new point layer with Menu Processing / Toolbox / Geometry by expression and this expression, based on input layer B:

array_first (
    overlay_nearest( 
        'layer_A', 
        $geometry
    )
)

Than connect the points using Menu Processing / Toolbox / Points to path` and from these path create a polygon with Menu Vector / Geometry Tools / Lines to polygons.