Acetylcholinesterase is an enzyme involved in the development of Alzheimer's disease. Symptoms of Alzheimer's disease are usually treated with acetylcholinesterase inhibitors or acetyl and butyrylcholinesterase inhibitors. So, if you test your extracts against this enzyme, you will evaluate their inhibitory capacity against this enzyme.
When you test your extracts against oxygen and nitrogen reactive species, you are evaluating their antioxidant activity, which means, their capacity to inhibit oxidative stress.
It all depends on the purpose of your study. You can test your extracts in many different assays, such as, against different enzymes related with different diseases.
Acetylcholinesterase is an enzyme which is responsible for metabolism of ACh so the drugs decreasing activity/inhibiting AChE can increase the level of ACh and so can be useful in conditions in which ACh activity is decreased Ex.: Alzheimer's disease, Myasthenia gravis, etc.
But if you are focusing on Anti-oxidant activity then such agents can be useful in useful in neurodegenerative diseases like Alzheimer's disease, Parkinsonism and other neurodegenerative diseases, etc...
One more thing I would like to add that according to many research work, it has been proved that Phenolic constituents have anti-oxidant activity so your isolated compound can also show anti-oxidant activity...