I believe that the KVS tool sources one version of its information from Flightstats, in a much more convenient way; and in my experience Expertflyer's availability data when looking at a UK point of sale is not always particularly accurate, quoting some flights as sold out even for waitlisting ('C') when there is availability.
But I'm sure one or both will be along soon to correct these statements.
KVS does not source from Flightstats, however both use a US POS.
ExpertFlyer has the option for US or UK POS.
If you are booking flights from non-US or non-UK POS locations, the availability may be different again.
Personally I have never seen BA.com not match exactly what ExpertFlyer shows for availability.