This question already has an answer here:
In ArcMap, is it possible to automatically (through a tool) snap points in one point layer (A), to points in another point layer (B)?
I do not wish to snap the points to a line, nor do I wish to merge the points - I would like the end result to be a new layer (C) containing points from layer A that have been moved to match the nearest point in layer B. If a separate layer cannot be created, but layer A is altered to match the locations of layer B, that is also fine. I don't think a spatial join works for me, as far as I understand it, as I want the physical location of one of the point layers to move.
I also do not wish to merge the two sets of points, but keep them separate.