Diving right in:
The basis for conflict is found in the first section of the Rules for Submissions (RfS) devoted to General Principles. RfS I.3.a. reads in part
Conflicting Claims - A ... piece of armory that creates a false impression of the identity of the submitter will not be registered. Someone may not claim to be another, either directly by using a ... [piece of] armory that is identical to another's, or by unmistakably claiming close relationship to an individual who is in fact unrelated.
The key thought is that armory conflicts if it is identical to or claims close relationship to previously registered armory. What conflict is not is when two coats look like each other. That sometimes comes into play, but that is not the heart of the matter. This is why we are unsympathetic to the cry that the two coats don't look alike: we never said that they did.
"Close relationship" is considered as one cadency step. Cadency is the process by which younger (i.e. "cadet") branches of a family difference the original coat. In other words, the number one son will eventually inherit his father's coat without changes, but (in most of Europe) the younger sons will inherit their father's coat each in some slightly changed form. Standard modern heraldry texts will typically give a neatly systematized scheme of cadency, but in actual fact it historically has been far messier.
The conflict rules, then, are really an attempt at defining (reasonably) objectively "what is a cadency step". More accurately, they try to define "more than one cadency step". Clearly one way of achieving this is to have two (or more) cadency steps: in rules jargon this is what we mean when we talk about two (or more) CDs. Other larger changes weren't used for cadency, so clearly no inappropriate claim is being made. This is what we mean when we talk about addition or change of primary charge.
To close this first part with an historical note, this standard is not that different from that established by Richard II in in 1390 in the famous case of Scrope v Grosvenor, and is fairly close to that used by the English College of Arms today. There are some SCA circles in which conflict as we practice it is very controversial. I am not without sympathy for that position. I would be the last to suggest that the historical record mandates our checking conflict. (Our ancestors were inconsistent in enforcement: Scrope v Grosvenor stands out for its exceptional nature as much as anything else.) If, however, we stipulate for this discussion (and please, everybody: lets do!) that the SCA is unlikely to change the policy of counting conflict, it is well that we follow historical ideas of conflict as much as is possible. In this I think we do a reasonably good job.
A second historical note, of SCA history this time, is that cadency is not the only way we have ever done this. The Old Rules considered conflict to be a visual matter, and attempted to quantify visual similarities. They were an impressive edifice, but horrendously complex. Those were the days of heraldic calculus as Talmudic disputation: fun at the time, but the New Rules (instituted some seven years ago) are much simpler. For those of you learning the rules, be glad you live in these modern times!