Why discuss lumens for their effectiveness on aquarium plants? Lumens is meaningless for plants. Plants do not utilize green light for photosynthesis. A higher lumen rating at the same wattage often means greener light. Lumen is a rating weighted entirely towards human perception. It has little to do with the value of a light for either growing or viewing plants.
Lux is lumens/square meter, so they are similar. They are both defined in terms that are meaningful to human perception of light Ė not plants. They stress the amount of energy in the green band to which humans are most sensitive Ė not plants.
Lumen is a measure of flux, or how much light energy a light source emits (per unit time). The lumen measure does not include all the energy the source emits, but just the energy with wavelengths capable of affecting the human eye. Thus the lumen measure is defined in such a way as to be weighted by the (bright-adapted) human eye spectral sensitivity.
The standard measure that quantifies the energy available for photosynthesis is "Photosynthetic Active Radiation" (aka "Photosynthetic Available Radiation") or PAR. This is blue and red light. This is why it is so important to get the spectral output of a bulb before deciding if is a 'good plant light'. You may need to add/mix bulbs to get a lighting that has good visual effects for the human eye and proper light for plants; because 'plant bulbs' tend to be purplish.