In relation to light and illumination, there is strong difference. Don't forget many languages do not have really equivalent terms for the samne things, so vocabulary authors tend to use the nearest one they think is correct. But as these people are not experts, they could be frequently wrong (nobody could be an expert for everything, so it should not be seen as their fault, but you should always anticipate such kind of errors...) E.g. in Czech language there is no really equivalent term for an "efficacy" either, even when many people very frequently use the incorrect "ucinnost" (what mean "efficiency")
"Efficiency" means a ratio of the output vs inputs expressed in exactly the same units, with exactly the same physical meaning. That include obvious power transfer efficiency (W/W), energy efficiency (J/J), quantum efficiency (#particles/#particles) and so on. As it is a ratiometric measure, it is frequently presented in units like percents, ppm,...
The "Efficacy" mean a ratio of values of not directly comparable metric, like the lm/W.
So when speaking about light sources, one light source could have an efficiency of e.g. 20% (so waste 80% of the input power as heat), but an efficacy of 110lm/W, while another light source could have the efficiency 30% (so waste only 70% as heat) and efficacy of only 90lm/W. So from these two, the one with higher efficacy (so generating more light) is the one with lower efficiency (so the output power is lower).
The luminous flux is not the same as radiated power. They are related via the spectral sensitivity curve and the spectrum of the light, but not the same. So when the first radiate only in the spectrum part just around where the eye is most sensitive, it generate a lot of lumens with only small radiated power, spo even when it's efficiency (radiated power / input power) is low, it's efficacy (luminous flux / input power) could be still high, compare to a source, what radiate a lot of power on wavelengths, where the eye is not as much sensitive (e.g. to get good color rendering)
Exactly this is the reason, why the full spectrum (CRI >90) fluorescents have lower efficacy than CRI80 tri-phosphor, even whenn they have about the same energy efficiency: The high efficacy tri-phosphors sacrifice some color rendering just to not spent power on wavelengths, where it does not contribute to the lumen output (mainly the red), while to reach the high color rendering quality you have to radiate across the complete visible spectrum, include the deep red, where the radiated power practically does not count for the lumen output.
And this is way more important with LED's, as there is much more freedom in spectrum design to optimize the color quality vs efficacy. And it is the way, how some LED's could reach high efficacy figures without being extraordinarily energy efficient (that is, where the main technology advance happen). So some cheaper maker could easily reach the efficacy figure of high-tech brand product: The cheaper one concentrate in his design all the radiated power into the spectrum, where the eye is most sensitive, so does not need as high energy efficiency to reach the desired efficacy figure. The difference is, the real hi-tech brand reaches that efficacy with better color rendering...
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