On the surface it looks like both libraries are designed to preform the same functionality with similar "drop-in" capabilities. Are there differences that I need to take into account when making a decision about which to use? If so, what are they?
I would not use ESAPI-PHP. It's a port of the Java ESAPI framework, (which has had its own issues, but soldiers on, more or less, mostly less) but the PHP port has been dead for a considerable amount of time now. I wouldn't trust it as far as I could throw it.
I can't speak authoritatively to the PHP Security Project, but at a glance, the PHP Security Project may be in a similar state (their project page and site announces their upcoming participation in AppSecUSA 2013, which happened over a year ago) but at least it was a native PHP framework designed for PHP, so there may still be use to be had from it. It's potentially worth researching, at minimum.