The Robe was the seamless, woven in one piece, tunic worn by Jesus that the Centurions cast lots for as Jeses hung dying on the cross. It was customary for the executioners to divide the belongings of the executed amongst themselves.
Cloth was one of the more valuable products of ancient industry with each piece representing many man-hours of labor. Up until the industrial revolution, textiles were among an estates most valuable possessions.
The centurions, seeing that this particular tunic was not sewn but woven as a single piece (perhaps knitted??) The decided that instead of cutting it into parts to share equally, they would gamble for it. Who won it went unrecorded.
The Shroud is a bolt of fine linen (twill, three over one) woven in one complete job and could have taken an expert weaver several days of labor to produce. The quality of this cloth would have been suitable for a very wealthy man. It was, after all bought by one, a member of the elite of Jerusalem, the Sanhedrin. The shroud was used, along with at least one other smaller piece of linen, as a burial cloth for Jesus.