It also needs a buffer, so all bets are off.
In general yes, however you still have the advantage of the 3dB higher input signal, so at least noise should be 3dB down, unless the noise of the input stage is higher already or the 3dB gain is lost by gain difference.
The last seems to be the case.. the input voltage of both is exactly the same. Which at least means they do handle the two signals differently. Since output is equal with equal input, the balanced input must be 3dB down in gain, explaining no gains in SNR.