Windows XP Network Printing?

A network with several Windows XP machines.

Computer A is directly connected to an HP printer. Computer B is connected with Computer A via a network.

Computer B sees the printer in the network, adds it and confirms that the printer is in ready state. It adds the printer with both Active Directory and directly (\\ComputerA) (“Add to Active Directory” ticks changed in printer settings).

When Computer B prints a document, a message shows up saying the job is complete, however the physical printer does nothing! Printer prints Computer A’s tasks with no problem.

What can be the cause?