Q. Using APA format, do the appendices come before references or vice versa?

Usually with a question like this I like to answer with a smoking gun, i.e. on p. 4014 of the Publication Manual of the American Psychological Association it states "whatever you do, make sure you place your appendices after the reference list!," but no such luck with this question.

However, there is a circumstantial case that the appendices should go after the reference list.

a). If you follow the progression of chapter 2 of the Publication Manual of the American Psychological Association, 6th edition, "Manuscript Structure and Content," it moves sequentially through the parts of the paper. The end matter, in this order, consists of -

References
Footnotes
Appendices and Supplemental Materials

The sample paper does not have appendices, but it does have footnotes and they appear after the references.

b). I checked the APA website, hoping it might provide an answer. Under a search for appendices, I did find this vague reference - http://supp.apa.org/style/pubman-ch02.13.pdf

It leads to this published article - 

Grossberg, S., & Pearson, L. R. (2008). Laminar cortical dynamics of cognitive and motor working memory, sequence learning and performance: Toward a unified theory of how the cerebral cortex works. Psychological Review, 115 (3), 677-732. doi:10.1037/a0012618

You can view it here - 

https://akin.css.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=psyh&AN=2008-09896-006&site=ehost-live&scope=site

In this article from Psychological Review the appendices appear after the reference list.

c). Also, this is the "spirit" of the appendix from the Publication Manual of the American Psychological Association, 6th edition - "Sometimes, material that supplements article content would be distracting or inappropriate in the body of a manuscript. Materials of this type can often be included in an appendix ... (p. 38).

I would argue that the reference list is much more important to the body of a manuscript. You can't have an article without it, but you can have an article without an appendix. Therefore, the reference list is first, and the appendix next as a supplement.

In your question you imply that somehow you have found in the APA manual an example of the appendix coming before the reference list. If so, I was not able to find it.