According to Bottlehead & co, the CCS can offer more gain with a given tube, as it acts almost as an "infinite" plate resistance, but at the same time, it sets the tube's operating plate current, as would a normal value plate resistor. The BH typpical driver for a 2A3 or 300B is one 12AT7 / ECC81 section, set around 3.5 mA with a CCS in the plate circuit. The other triode section is unused in some cases, or employed in a shunt regulator, in others.
Some folks like the CCS sound, and some just plain don't... I've used it (in a few Bottlehead amp kit builds) and liked the results, but as with anything, it does increase the circuitry's complexity and can decrease its reliability.
Some say that if one surrounds a tube with transistors, then one ends up with a "transistor sound" Yellow_Light_Colorz_PDT_02
The one case where a CCS reaally "shines" is when it acting as the "tail" resistor in a Tong Tail Phase Splitter. This can set the current and not require the high negative voltage ssupply for the normally high-value "tail" resistor.
Another "Trick" is to use an LED as a cathode resistor, there is some discussion of this in the G*S*G topic. Generally a red LED is used, and the tube's cathode voltage has to be around 1.6 to 1.8VDC above ground (zero volts) . LEDs have a fairly low effective resistance, and do not need capacitor bypassing.
HTH
/ed B in NH (still waking up on this rainy cold first day of "spring"...