How is the Early voltage defined?
I think you'll find that there is a small difference due to voltage-dependent reverse leakage of the C-B junction, but it's not significant under normal conditions. To a first approximation the leakage is independent of voltage (for Vcb > a few Vt).

If the transistor is being operated at very low collector current (nA) and high collector voltage the constant base current method will give you a measurably lower value for Va. 

I believe that constant Vbe is a better representation of behavior in real situations such as current mirrors.