These are confusing terms, I had to double check myself to make sure I give you the right answer. First, the definitions:
efficient - achieving maximum productivity with minimum wasted effort or expense
efficiency - the state or quality of being efficient
efficacy - the ability to produce a desired or intended result
effectiveness - the degree to which something is successful in producing a desired result
Efficacy is probably the least used of these words and effectiveness the most common one. When something is effective, it means that it works, that it does what it's supposed to do. It doesn't necessarily mean it's efficient, it doesn't mean it does it in the fastest time but it gives the results. Maybe we could say that going to a psychiatrist is effective, it works, it improves one's mental state, but taking a mood altering pill is much more efficient, it's fast (please, no comments about the actual therapy and medical side effects, I'm just trying to make a clear point - thank you).
"The opposition of capital punishment derives mainly from its inefficacy" means that those who oppose it do so based on the fact that it doesn't produce the intended results. If you used "inefficacy" instead, it would mean it doesn't give the results quickly or at a lower cost, for example. The opposition is arguing that it doesn't work, period: there is no discussion about the quality of the work. According to them, there is no result, it's not that the result is expensive and slow.
In terms of time, a car is much more effective than a bicycle. In terms of impact on the environment, it's the other way around. Which one is more efficient? It depends whether you're talking about money, fuel or physical strain.
The efficacy of a blanket to make you warm is pretty high but of a red pencil for that same purpose is extremely low.
I hope this helps clarify it for you.