Appears to be some confusion in this thread.
First, I have no specific 777 knowledge so can only talk generics .. however, the thread subject will only differ between Types by virtue of weights in the AFM .. the rules are the same
Second, I would recommend heeding HN39's counsel.
The certification (design) requirements include, amongst the various WAT limits ...
(a) approach climb WAT. Addresses the missed approach starting OEI, gear up, approach flap, etc.
(b) landing climb WAT. Addresses the missed approach starting AEO, gear down, landing flap, etc.
Both of these limits must be observed (ie whichever might be the more restrictive is critical). Note that the required gradients are not the same. As to why the OP's weights differ would require access to the source documentation ...
Looking through the thread, the pertinent concerns appear to be ..
(a) Dispatch performance and Inflight performance usually have different factors
Think, perhaps, in terms that despatch requires full adherence to whatever the rules might be. Inflight we are looking at abnormals and emergencies and the priority is getting the bird back safely. Same approach with MEL. 8che, I think, made a similar observation.
(b) This is to of course help dispatch and fly with the gear locked down for technical reasons etc.
Pertinent MEL point. If you are INTENDING to commence a flight you must observe all the usual rules ergo, if the gear is down, you will need weight penalties to meet those requirements normally related to gear up.
(c) but the landing climb limit weight is also calculated with gear down
.. however, the approach climb is gear up. Therefore, for an INTENTIONAL despatch, the operator will need to meet the OEI approach climb requirement with gear DOWN.
(d) Gear down performance means gear cannot be retracted
Suggest no .. the landing climb requirement is to ensure that the missed approach can be initiated in the landing configuration until such time as the aircraft can be reconfigured to the (approach climb) normal missed approach configuration. I presume the 777 missed approach procedure from late final is something like .. thrust up, pitch up, flap approach, positive climb, gear up ... like other aircraft ? Thus, if you can't retract, you need to achieve appropriate gear up performance with gear down.
(e) a landing climb assumes you retract the gear while going around..
Common pilot misunderstanding .. the certification rules and the routine operating rules don't always see eye to eye .. similar case is stall recovery protocols.
(f) Approach climb is not an issue here
.. but it is .. the operation must meet both limits to provide for the OEI case.
(g) why don't they just say that in the landing gear non-normal procedure?
There's no real need to do so .. the weight is the lower of approach climb and landing climb ..
(h) because only the landing climb weight is used when performing the overweight landing cx list
That would make no sense at all as it ONLY addresses the AEO case.
(i) inferring that on this aircraft landing climb weight limit is always a greater consideration than approach climb limit.
That may be true .. would take an AFM to research.