I'm pretty sure it was checksum, not binary. Since I didn't care about the difference in case, which is the primary difference between the two.

select checksum('A'),checksum('a'), binary_checksum('A'), binary_checksum('a');

Results:

142 142 65 97

With what I was doing, case differences didn't matter, only actual content differences. That's the decision that determines which you should use. Of course, it's also critical that you use the same one on both sides of the equation.