Can voltages be different if watts are the same? Does DC vs. AC change this?
Just to correct your water analogy:

The Voltage is the water pressure,
The Current is the flow
The multiplication of Voltage and Current is the power and it is the same with water, the product of flow and pressure is the hydraulic power: 

If you have a pump running in a pool of water just recirculating you have flow (current) but no pressure (Voltage) and no power. It is not doing any useful work.

If you have a pump running with a valve on its outlet that is closed there is pressure (Voltage), behind the valve, but no flow (current) and no power. Again, no useful work. 

It is only when you have flow against a pressure that useful work gets done. Like when you pump water up to a high mounted tank, no so high that the pump can't do it!

To get a  clear picture electrically: A 1.5 V dry cell battery sitting on the desk has a voltage between its terminals but no current flow, since there is no circuit for the current to flow in. There is no power - no heat or anything. Power = 1.5 x 0 = 0.

Connect a 0.5 W 1.5 V 0.33 A light globe and current flows and there is power! Since the wire in the globe is the right size to carry 0.33 A at 1.5 V, the Power is 1.5 x 0.33 = 0.5 W and it gives out light (and heat) the total of which is 0.5 W.

The water flow analogy can even be used for alternating current but that is another story...