This invention relates to a rubber composition and tire having at least one component of a rubber composition comprised of at least one diene-based elastomer, to the exclusion of butyl type elastomers, which contains an internal dispersion of corncob granules. Such tire component may be, for example, a tire tread and / or cord reinforced rubber ply and / or belt. A tire tread intended for winter driving, although not necessarily limited to winter driving, is provided of a rubber composition containing at least one diene-based sulfur vulcanizable elastomer having a Tg below −30° C. and an internal dispersion therein of corncob granules. The running surface of the tire tread is configured with a combination of micro-protrusions of corncob granules from said corncob granule internal dispersion and micro-cavities therein created by a release of a portion of the protruded corncob granules during the abrading, or wearing, of the tire tread running surface as the tire is being run. At least a portion of said corncob granular micro-protrusions may be covered by a relatively thin membrane of tread rubber. The corncob granules may be colored with a suitable colorant of a non-black color to contrast with the tire tread color to better visualize their presence on the tread running surface. The tire tread rubber composition may optionally also contain a starch / plasticizer composite, sorbitan ester, methylene donor compound, methylene acceptor compound or combination of methylene donor and acceptor compounds, and may be reinforced with carbon black, synthetic amorphous silica aggregates or their combination.