I am making the distinction. However what you choose as an axiom in your science of morality is still a choice. You have to take what your moral instincts tell you and encode that instinct into an axiom. I can frame what I said from that perspective to help you understand.

Consider the situation I talked about and the two possible moral axioms that arise from it.

The psychopath that sat and watched the baby drown. Should he deserve any form of punishment? What should we encode as an axiom from our instinctual response to this situation? Should we just let people who sit by the pool watching the baby die not be punished? Is that the right thing to do?

What about the people who didn't donate a dime to Africa? Should they be punished? This is basically most people in the US. Should we punish everyone in the US?

Axiomatically speaking if one is punished so should the other.

This is the axiomatic conundrum. We evolved moral emotions to determine our behavior. Even when we try to interprete these emotions from an axiomatic perspective the conondrums remain on which axioms to choose. Depending on the situation neither axiom completely fits our interpretation of what is right or wrong.

Some people resolve this by choosing not to punish anyone by basing their decision on the axiom that punishment should only be reserved for murder by action not murder by inaction. However that is just an axiom you arbitrarily chose. Emotionally speaking murder by deliberate inaction is something everyone feels is fundamentally wrong as well and can be shaped into an axiom. The problem is this axiom conflicts with our instincts that are telling us that the people who killed starving kids in Africa by refusing to donate money are not guilty of murder.

There are other moral situations that can hijack our brain. Consider a train that is about to run over and kill 5 people tied to the track. You can save those 5 people by pulling a switch causing the track branch to switch to another lane. This lane only has one person tied to the track. Do you pull the switch? Save 5 people or save 1 person? By default the train will run over 5 people.

Think about your decision before I adjust the parameters. Most people choose to save 5 people over saving 1 person and they will flip the switch.

Imagine the same train. The track is now linear with no branch. 5 people are tied to the track. You see a really big fat man standing next to the track. You can save all 5 people by sneakily pushing the fatman onto the track. The fatman will die but his weight will stop the train saving the 5 people. What do you do?

The situation is the same, only superficial parameters have changed.

Basically the situation is this: you have the choice of saving the lives of 5 people or 1 person. Axiomatically we by default think that you should save 5 people over 1 person.

However while both situations are the same we feel the the second situation tends to feel wrong. It feels like murder if you save 5 people. The axiom of murder by action and the axiom of saving 5 people over 1 person in this situation form a moral paradox and the feeling of what's wrong or what's right changes depending on if we push a person onto the tracks or if we flick a switch.

This indicates that when you try to encode our moral instincts into axioms they form inconsistencies. An inconsistent system means the system is arbitrary and made up. It is not a "science" and just an arbitrary set of inconsistent behavioral rules. In other words, it's just a biological group of behaviors that coincidentally helps humans survive in a society; it is not some universal truth in the universe, not like the laws of physics.

Is it worth studying an arbitrary set of rules as if they were some universal truth? No. I think it's excessive. You need to create laws so that society functions but any analysis deeper then that (like the philosophy of ethics for example) is a pointless endeavor. There is no axiomatic or fundamental truth here.