The water remains the same in either case, and each will quench physical thirst as well as the other. The difference is that giving that cold water in Jesus name, makes the person receiving the water more aware of other thirsts that need quenching, specifically spiritual thirst, and begins to point the person to where those thirsts also can be satisfied i.e. Jesus. It also indicates that the person giving the water is aware that his motivation is not only human kindness, social awareness, guilt, or any of the other human motivations, some good, some less so, that may motivate such a kind deed, but that he is giving the water as part of his vocation as a disciple of Jesus.
Reminds me of the old "give a man a fish" saying. Give a man a fish and you fed him for a meal, teach a man to fish and he'll be out of your hair for hours on end. More completely, compare to Jesus' conversation with the woman at the well. He asked for water to quench His thirst. When she wondered that He a Jew would ask her a Samaritan woman for water, He directed her to contemplate that He could give her living water (Himself) the would quench a thirst for eternity. By giving the water in Jesus name, we also offer living water.