What is the difference between a protein and a factor?
Short Answer

There is no agreed upon naming convention for proteins - there are some rough standards because in language people usually try to convey their ideas in a way others can understand, but that doesn't necessarily mean fixed rules.

Longer Answer

I think it's important to recognize the process for understanding what proteins do is not always straightforward. Most often, a protein's function is first understood by seeing what happens if that protein is absent (and sometimes overexpressed).

The terminology factor implies that a protein is either modulating a process or at least is not by itself sufficient for a process. That is, it is named because when it was omitted, some other measurable process changed. It  maybe implies something that has its main function by binding, rather than catalyzing a chemical reaction, or at least that the actual mechanism of action is not yet known at the time of naming.

You would not expect, for example, an enzyme with a known target to be called a factor: it's most likely to be named according to its target and the type of reaction being catalyzed.

However, it's possible for a name to stick from when there is less complete understanding. Something might be called a factor initially because of how it influences some process, but the actual contribution is only understood later. Just for an example, take Complement factor I. Initially named because it had some role in the complement system, now it's known that it enzymatically cleaves another protein.

Importantly, there isn't any pure terminology here or consistent naming convention: in most cases, it just goes back to however the first person to describe the protein discussed it. Some proteins are named (controversially, I'll add) things like Sonic hedgehog - related to the similarly named Hedgehog signalling pathway which is all named because of one person/group's creative description of a related fruit fly phenotype.

Naming something "protein" just identifies that it's a protein, little more.