What are the pro's and con's of using PostgreSQL vs MySQL for binary data? I read that MySQL has binary safe blobs, but PostgreSQL does not.

So far what I've found suggests that "binary safe" is something that really relates to functions and handling of data, not the data itself. It seems that the difference is that MySQL natively has ways of storing and manipulating blobs without having to specify data formatting, but PostgreSQL requires using either "quoted-printable
encodings" or "base64" encoding for binary data.

Is that all there is to it? What else should I worry about with respect to the differences between MySQL and PostgreSQL in the handling of blobs? I'm starting a new project, not migrating data, so I'm just wondering if there are particular reasons to choose one or the other with respect to their handling of binary blobs.
That reference you link to is complete nonsense.  PostgreSQL supports binary data and binary-safe handling of data just fine.

There will be some small differences between MySQL and PostgreSQL about how the types and certain functions are named, but overall it won't make much difference.  There are probably other, better reasons for choosing one system over the other.