I have a lamp that is rated at 60 Watts for an incandescent bulb with a disclaimer to use a 13 Watt bulb instead when using a CFL replacement. Do I really need to respect that disclaimer? Would not a 23 Watt CFL be just as safe to use? It still draws less power than a 60 Watt incandescent, right?
As far as I know, the only risk at issue here is of overheating of the bulb. The heat comes from two sources: the output and waste heat of the bulb, and from resistive heating of the fixture wires and switch and socket contacts. CFL bulbs draw less current and produce less heat for a given light output; however, they are far less tolerant of high temperatures!
Therefore, the rationale of the warning label may well be to warn that the fixture is not sufficiently ventilated for a CFL bulb above that power, even if the fixture is itself capable of surviving that amount of heat.
However, it is also quite possible that someone converted the "equivalent wattage" (which is a statement of light output and therefore almost completely irrelevant to power/heating) without thinking about it. I recently (2016) saw a lamp which had a claimed limit like "60W incandescent, 13W CFL, 7W LED". That's clearly just converting, and it's not going to be accurate for the wide variety of LED bulb types available.
The issue here is light output, specified in lumens. An incandescent bulb puts out about 15 lumens per watt, so a 60 Watt bulb will put out about 900 lumens of light. However, a CFC may put out 45 - 60 lumens per watt (3 - 4 times as much), depending on the design. They are simply a lot more efficient at converting electrical energy into light. Because of this, a 13 Watt CFC will put out the same amount of light as a 60 Watt incandescent bulb. Putting in a bigger CFC simply gives you a lot more light.
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Would not a 23 watt CFL be just as safe to use? It's still drawing less power than a 60 watt incandescent, right?
The disclaimer probably would be for the light intensity...probably its a reading lamp, so higher intensity light can damage your eyes...
Mind you this is just a speculation!
I'm playing a bit fast and loose with the units here -- Watts are a measure of power, but the idea should be clear.
Typically, you find these ratings on the socket where there is likely to be a shade in use. The smaller the lamp, the more likely you are to use a small shade, and the closer the shade will be to the actual bulb. The warning is there so you don't set the shade on fire, or melt it. 90% of the power (watts) of a light bulb is emitted as heat. wikipedia. So, a 60 watt bulb emits 54 watts of heat, and 6 watts of light. A CFL uses less power for the same light by a factor of 3-4. meaning a 13 watt CFL will emit the same 6 watts of light -- giving off 7 watts of heat.
TLDR; The wattage rating is for fire safety, you could use a 60W CFL in the socket, if you could find one of appropriate size. The rating has nothing to do with the amount of light produced. Only heat.
There is one issue I see not addressed by other answers, and that is, a 23W CFL is not appropriate in any kind of lamp that has a dimmer unless it is specifically a dimmable CFL bulb. It's quite easy to accidentally use a lamp with a dimmer and a CFL that isn't designed to be dimmable, and this is a very important thing to avoid because it can cause overheating and even a fire.
The reason is due to the way a dimmer works, essentially by switching the power to the bulb on and off many times per second on some kind of duty cycle approximating the amount of dimming required. A CFL bulb will use several times more than standard operating current during startup, and this can lead to overheating and a fire, or at the very least drastically reduced life for the bulb.
Depending on the type of lamp fixture you are using, it is possible there may be a similar thing going on even if there is no dimmer. For instance, an older garage door lamp I had turned out to be incompatible with CFL bulbs due to the use of a SCR like component that effectively prevented the bulb from coming on properly some of the time. With incandescent bulbs effectively being phased out newer devices are of course being designed with CFL (and LED) bulbs in mind, but it is always important to consider whether an older lamp is compatible with a CFL.