I had always wondered why these two types of tubes existed, with such similar characteristics. Apparently the 6C5 came first, and the 6J5 is an improved version.
From a
European Sylvania tube application data sheet:
...In general, the applications and operating conditions are similar to those of tubes like the 76, 37, 6C5 and 6C5G...
Although the 6J5 and 6J5G have the same amplification factor as the 6C5 and 6C5G, the mutual conductance has been increased, with a consequential reduction in plate resistance. The output capacitance of the 6J5 is about a third that of the 6C5, and the design of the tube is such that the 6J5G is especially adapted to function in equipment at very high frequencies.