Same pinout, but NOT the same tube
The voltage drop is very different between a GZ37 and GZ34, so the bias point on the power tubes will be different when using either one. Every rectifier type will either raise or lower the bias voltage depending on the rectifier's voltage drop. This will change the sound, but also, may be over extending your power tubes, leading to there premature death. The GZ37 has a greater current draw (3 amp) than the GZ34 (2 amp), and the voltage drop is about 10 to 20 V greater than a GZ34. If your transformer allows for 3 amps of heater current at 5 v then the gz37 should work giving different output voltage and load characteristics. If the power stage of the amp is self-biased, it will settle itself with either tube... The sonic difference is primarily due to placing the bias at a different point.
Still the specific issue of subbing a gz37 or 5u4 rectifier in a amplifier that has 3 amp capability has been the source of much debate with at least one person I know experienced in the subject. The bottom line: it's best to use the tube that your circuit was designed for and when deviating from that standard to proceed with caution. As you can either prematurely waste a tube or smoke a transformer.
MikE