The invention relates to a tire with a tread of a cap/base construction comprised of a two-ply (dual layered) tread cap rubber layer comprised of a combination of an outer tread cap layer and an inner tread cap layer together with an underlying tread base rubber layer which underlies the two-ply tread cap layer. The tread cap layer is of a lug and groove configuration with at least a portion of the grooves extending through the outer tread cap layer into the inner tread cap layer without extending to the tread base rubber layer. The outer tread cap rubber layer is comprised of a carbon black-rich rubber composition to promote dry traction for the tread running surface. The inner tread cap layer is comprised of a softer silica-rich rubber composition to promote wet traction for tread running surface as the outer tread cap layer wears away to expose the softer inner tread cap layer.