When should I use APCI over ESI?

First it is best to understand the differences in the techniques. ESI ionisation occurs in the mobile phase with the addition of either a basic or acidic modifier to promote ionisation (commonly forming either M+H or M-H ions)
APCI ionisation occurs at the corona needle via a cascade reaction involving gas-phase ion-molecule reactions.
ESI ionisation is the most senstivive technique wrt to SN for polar analytes whereas APCI is the most sensitive for non-polar analytes.
APCI can be a useful technique for overcoming matrix effects (competitive ionisation) and sometimes where high LC flow rates are required can achieve improved linearity over ESI.