There are Hadeeths which indicate that Zamzam water is noble and blessed. In a Hadeeth confirmed in the Sahih, the Prophet (sallAllaahu `alayhi wa sallam) said about Zamzam water:
إنها مباركة إنها طعام طعمة
"Indeed it is blessed, and that like food, it fills." (Muslim no. 2473)
And in Abu Dawud's narration, there is this addition:
"And a cure for illness." (Abu Dawud At-Tiyalsi 2nd Volume no. 61)
So this Hadeeth proves its virtues, that it is a filling food, a cure for illness, and that it is blessed.
The Sunnah is to drink it as the Prophet (sallAllaahu `alayhi wa sallam) drank it, but it is also permissible to make ablution with it, to use it to purify yourself after you have relieved yourself, and if necessary, to take a shower using it when you are Junub (impure).
It is confirmed that on one occasion, water poured forth from between the Prophet's fingers, and then people took what they needed from that water, using it to drink, to make ablution, to wash their clothes, or to purify themselves after relieving themselves.
Zamzam water may not be the same as the water that poured forth from between the Prophet's fingers, but it is also not more blessed, for each has its place as pure water. If it is permissible to use the water that miraculously poured forth from between the Prophet's fingers for all the purposes mentioned above, then it is also permissible in the case of Zamzam.