"Significant results" vs "Key outcomes or Other achievements" The NSF annual report asks for "Significant results" and "Key outcomes or Other achievements." What is the difference between these two categories? 
 A: These two categories, plus two more, are not mutually exclusive. For some research one category may be better applicable than another, and in any case you are only required to provide information for one of the four ways that describe what you have accomplished to reach the goal of your project.
1. Major Activities; 2. Specific Objectives; 3. Significant Results; 4. Key Outcomes or Other Achievements
As a mathematician, I presume the goal of the project is to produce significant results, say, prove a theorem, so typically you will address #3. In other branches of science the goal may be less specifically tied to concrete results, and for example involve a medical trial or a field excavation covered by #1.   
Best practice could be to list under #3 the theorems proven, or steps taken towards the proof of a theorem, and under #4 then discuss the broader implications.     
Item #4 also explicitly asks to discuss whether there are goals which you have not been able to meet, while #3 asks for "negative conclusions". These two questions are also somewhat overlapping: I would include failed attempts towards a specific result under #3, but if you decide that an entire objective will not work out, #4 would be more appropriate.
