Objective evidence refers to data that can be seen, touched, tasted, smelled or heard. Any evidence that can be perceived by the five senses is objective evidence. A medical doctor collects objective evidence for diagnosing her patient by checking his pulse, taking his temperature and looking at his blood through a microscope. A police officer takes fingerprints and photographs the physical condition of the crime scene. An auto mechanic looks under the hood at the engine and listens to the sounds it makes. The knowledge each investigator gains in these examples comes from objective evidence.
A narrative is a story and narrative evidence comes from stories. Medical doctors ask patients to tell them their family history, how they sleep at night or what sort of stress they experience during the day. Police offices interrogate witnesses and suspects to gather information about the crime. Auto mechanics ask car owners how the car has been running or if they've felt anything unusual in its performance.
Objective evidence is solid because it can be physically observed and shown to others if they are in doubt. Objective evidence is the tangible result of some cause and very often that cause can be explained by looking at the objective evidence. However, there is a possibility that objective evidence can mislead an investigator in mistaking the cause. Reality is very complex with many different forces working. This is especially true with subtle subjects, like psychology, which don't present objective evidence easily.
Narrative evidence can often go beyond the material empiricism of objective evidence because it includes factors like imagination, intuition and self-reflection. Narrative evidence can admit subjective strengths like accumulated life experience or wisdom into the process of investigating a cause which can often times transcend objective evidence. The disadvantage is that narrative evidence relies on human subjective faculties, like memory and opinion, that can often be distorted by emotion, leading to mistakes in identifying a cause.