Water use efficiency is about how much waste in the use, so if you achieve similar results with 1000 liters drip irrigation, 10000 liters spray irrigation, and 1000000 liters flood irrigation, the drip is the most efficient for the specific result. This is not to say that drip irrigation would be best and most efficient for every instance, such as flooding rice may be most efficient when producing the quantity of rice desired.
Water productivity is more about how much can be produced with a specific water quality. Low productivity water may have poor habitat relative to nutrients, temperature or other conditions for aquatic insects and fish. High productive waters have conditions suitable to produce more or relatively high yields. Same for rice, low productivity waters a poor or marginal - perhaps salinity or other issue.
It is however possible, for local or cultural differences in how some terms are used and applied.
Your interesting question has been discussed previously in ResearchGate. You may be able to find the answer to your question through the following link.
In the literature dating back to the 1960’s, the researchers make no distinction between the two terms (WUE and WP). But, the most recent paper on that subject (Heydari, 2014) indicated that WP is distinct from WUE as WP refers to crop production in relation to total water consumed while the WUE is a dimensionless ratio of total amount of water used to the total amount of water applied, “as WP terms are not dimensionless, i.e. cannot be categorized in efficiency terms, they are just some ratios with different units in the numerator and denominator”.
Heydari N. 2014. Water productivity in agriculture: challenges in concepts, terms and values. Irrig. and Drain. 63: 22–28.
Water use efficiency is the ratio of water used in plant metabolism to water lost by plant through transpiration while water productivity is crop yield per cubic meter of water consumption including rainfall water (green water) for rain fed areas and both rainfall water (green water) and diverted water from water systems (blue water) for irrigated areas. Please have a look at these links for more details.