When the pump is running, is water leaking from the steam wand even if the knob is closed? There is a spring-loaded valve in the brewhead that won't open unless there's enough pressure, so if your steam wand is leaking badly that could be the cause.
If your wand is not leaking then either 1) the above mentioned spring/valve is seized shut, 2) there is calcium buildup in the boiler blocking the water passage to the brew head, or 3) your pump is not producing enough pressure to open the valve.
1 and 2 can be verified by removing the brewhead screen again, removing the brass piece that it screws into (if you don't have a wide flat screwdriver then nickel should be the correct size) and examining the spring and valve (a small silicone nipple). With the spring and nipple out you should be able to shine a light into the hole and see into the boiler as well as poke a paperclip or dental pick into it. If you encounter any resistance or see a white chalky residue then it might be plugged with calcium and you should run a descaling cycle.
If the valve is functioning properly and the boiler is free of calcium but you still aren't getting any water from the brewhead then it most likely is your pump. You can purchase a replacement here: http://www.shop.partsguru.com/Barista-Rio-Vapore_c428.htm