The difference, Alyssa, is that they are two separate and distinct organizations with different goals although with an overlapping, to some degree, membership.
DSMs are the world of the psychiatric association and something you and everyone else should be aware of is that psychiatrists and Clinical Psychologists can legally write prescriptions, along with APNs. Why does that matter, say thee? Because the incestuous relationship between diagnosis and it's definitions as placed into DSM, decisions about what stays in, is added or goes and how treatment is viewed is very often quite intimately involved with making a new diagnosis that (here's the real surprise) that just happens to dovetail with the approval of a new drug one of the effects of which is to treat exactly the symptoms of the new disorder!!
Those who prescribe get preferential treament from pharmaceutical companies. As do those who make the 'bible" of the trade that uses the meds produced by said firms.
The other APA doesn't get those perks. I suspect that's one of the reasons you saw the call by the Psychological APA this past summer to change the way transfolk were dealt with. While there's some whispers right now that the Shrink-wrap APA may attempt to further pathologize us all by formalizing the Blanchard dichotomy. That would be the Blanchard who's on the committee that will be handling sexual disorders.
Nichole