GFCI circiutry constantly checks for a difference in current between what?

Current flowing through the "line" or power wire to the load(s), and current flowing "back" from the load(s) through the "neutral" wire.

This link provides a more detailed explanation:
What is a GFI/GFCI?

Daniel

The GFCI does detect the current difference between line and neutral. It does this with a current transformer where the line and load wires are wound in opposite directions. Therefore the subtraction is done in the current transformer.

It also detects a grounded neutral. This is does by modulating he hot and the neutral currents with opposite polarity. This would normally cancel. It cannot modulate a neutral connected to ground after the GFCI, so it detects the presence of the modulation.

240 v GFCI's are a bit different. You have 3 wires to contend with L1, L2 and N,