Both terms in fact correspond to different things.
BNF stands for "Biological Nitrogen Fixation" and accounts for all the nitrogen that is fixated by organisms to the soil and/or plants. If you're working with legumes, this generally refers to rhizobia, which are particular bacteria that can produce nodules and fix nitrogen from the atmosphere into them. Legumes can then use up that N that has been fixed from the atmosphere.
NUE (Nitrogen Use Efficiency) is the total amount of biomass that is produced per kg of N which is absorbed. Sometimes, there are variations to this term and instead of absorbed N some papers can refer to applied N. However this is rather confusing as there are many factors that can affect N absorption (particularly in non-irrigated experiments).
A legume has two N pools: one that comes from BNF and one that comes from the soil, BNF also has a negative relationship with the N content of the soil and so the more N your soil has (or the more you apply), the less BNF you will measure.
I'm currently working on a paper that deals with BNF and inoculation in lucerne (Alfalfa). Let me know if I can be of assistance with any other thing. Hope this helps!
The difference between BNF and NUE were already explained. I just wanted to suggest you the following reading, which describes how to calculate nutrient use efficiency. Hope this helps.
Jose' provided an excellent description of differences in BNF and soil N.
Nitrogen use efficiency refers to how efficiently plants use accumulated N (BNF and soil N) to produce biomass ( ie. N use efficiency = Total Biomass produced (kg) / Total N accumulated (kg). As soil N concentration increases the BFN contribution to plant N accumulation will decrease. I would not anticipate that this would impact the efficiency of nitrogen use in production of biomass.