Recently, I was explaining to my high school class what the Borda Count was. We had taken a class survey on something and everyone ranked their choices in order of preference. I calculated the Borda Count scores and showed the class the official class ranking of their choices. One student asked "That seemed a bit complicated...couldn't you have just taken the average ranking of each option?"
I went back and did her suggestion, and just looked at each choice's average ranking, and it turned out to lead to the same outcome as just doing the Borda Count. So now I don't know if this is equivalent to the Borda Count, or is there some obscure situation where these different methods lead to different societal ranking. Are they equivalent or is there some counter example? I feel like if they were just equivalent, then this would be the standard definition of the Borda Count process, right?
They are equivalent.
As pointed out in the comments, the lowest average rank always has the highest Borda score. If there are $N$ voters and $M$ candidates then a candidate with average rank $r$ will have a Borda score of exactly $N(M-r)$.
So why use the count over the average?
First off, note that neither is much easier to compute, as we can quickly compute either of these measures if we have the other, because they are in a direct relation.
Computationally the count is perhaps a little simpler. It allows you to work only with natural numbers, which is nice. Also, ranked preference ballots tend to allow a voter to leave alternatives unranked. If we want to compute an average we have to tally each unranked alternative as having rank $M$, whereas the count can simply ignore them (since they receive 0 points).
Personally I also find the count simpler to understand and explain than the average, but of course you may very well disagree.