Does a dish washer uses up more water or less than conventional hand wash?
The answer depends on the type of dishwasher that you have.
Modern dishwasher actually use less water than conventional hand-washing, on condition that you use a full load.
They are very efficient with regard to electricity usage too.
If you are trying to save water, I suggest that you take a look at http://funadvice.com/r/151jqdn9099
It depicts lots of practical water saving tips that can reduce your water bill by 30% or even more and save you a lot of money.
Too many variables to say for certain. You have to factor in whether or not the dishwasher is one of the newer ones that use less water, or if you use running water to rinse your dishes, or if you have to rinse your dishes before putting them into the dishwasher ... etc.
The dishwasher is the one efficent way to wash dishes, or so I've heard.