Joining attributes from nearest point in QGIS?

Using "Distance Matrix" you can calculate the nearest point. Make sure that both layers are in the same CRS! Then, you can load the results and use them to join table A and B.

In "Distance matrix" chose: "Linear distance matrix" and "use only the nearest k target points: 1"

Load the CSV using "Add Vector Layer" and join the layers in Layer A's layer properties - join tab.

And the result:

The NNJoin Plugin should do the job. For each feature of the input layer it adds all the attributes of the nearest feature in the join layer and also adds a distance attribute with the distance to this feature.

In QGIS 3.8 and above the "Join attributes by nearest" native tool in Processing Toolbox does the job easily!