JST to JST connector
Connecting these two circuits is likely not what you want to do.  The USB li-poly charger is designed to take 5VDC from the USB and provide charging current at a voltage appropriate for recharging a li-poly battery.  The lily pad supply is meant to take a li-poly cell and step its voltage output up to 5V with a switching regulator for use in wearable computing circuits.  

So first of all, it looks like you're taking 5V and ultimately converting it to 5V which is unnecessary.

Second, the USB charge controller isn't designed to provide a fixed voltage output, which is what the Lilypad converter expects.  The USB charge controller is going to treat the Lilypad board as if it is a lithium polymer battery,  attempting to charge it in stages by providing different voltages and currents based on characteristics of the load.  Not the ideal supply for your simple switching power supply.