I believe the "H" stood for "high brightness" or "high intensity", as it had an additional high voltage supply for the post-deflection accelerator.
The schematic here:
http://bama.edebris.com/manuals/dumont/304
shows the differences between the 304 and the 304H. The additional circuitry used in the 304H is shown in a dotted line area in the power supply. The note near the CRT intensifier connection describes the difference in the circuit.