I think the concepts are the same. As the Mad (Flt) Scientist neatly summarised above the difference only lies in the definitions.
The B737 charts I have to hand input 'Field Length Available' (I'm assuming this is TORA=TODA=ASDA) to output FLL TOM. A second set of tables take density altitude and TOM to derive a V1. The whole arrangement seems designed for simplicity at the possible expense of accuracy. The British balanced field charts of the '50s and '60s, not just the V bombers - I used them only as an example, are identical in concept. The only issue is the definition, the Americans refer to a balanced field as occurring when TODR=ASDR, the Brits when TODA=ASDA.