There's no difference between the taxes payable for arrivals into either airport - it's £0 in either case

But the key point is that BA has decided only to offer FCOD if you are changing to a different flight where the departure and arrival airports are exactly the same. That's arguably harsh and it certainly reduces the value of FCOD as a facility, but it's clearly set out in the conditions when buying the fare.

This policy might be due to market segmentation (e.g. LCY/LHR flights usually attract a premium over LGW ones), the constraints of poor IT, or something as bureaucratic as the fact that pricing and fares for LHR, LCY and LGW flights are managed separately by each operating company and they don't want to tread on each other's toes.

It seems most unlikely that it's due to differing airport fees, firstly since doesn't apply for inbound flights to London, and secondly because this doesn't seem to be an impediment to re-routing during irrops.