Maximum likelihood estimator of active time delay and passive time delay
Very intuitively, the Generalized Cross-Correlation is a "standard" cross-correlation of the windowed signals (I'll restrict myself to window-GCC, I'm pretty certain there's others, too!). Windowing happens to increase the "peakiness" of the cross-correlation.

It's basically down to the same trade-off between temporal and spectral precision that you consider when applying a window for the DFT. You trade energy in the "temporal edges" (read: start and end of the signal are attenuated by the window) for less leakage in the spectrum.

Now, back to your problem. In the active TDE definition you use, 

$$\begin{align}
R_{x_1x_2} =&amp; E\left\{x_1(t)x_2(t+\tau)\right\}\\
=&amp; E\left\{s(t)s(t+\tau)+ s(t)n_2(t+\tau)\right\}\tag1\\
=&amp;\underbrace{E\left\{s(t)s(t+\tau)\right\}}_\text{what contains the peak we're looking for}+ \underbrace{E\left\{s(t)n_2(t)\right\}}_\text{measurement noise}\tag2
\end{align}$$

Now, your GCC with a window $w$ looks something like (how $w$ is determined differs for different GCC approaches)

$$\begin{align}
\text{GCC}_{x_1x_2} =&amp; E\left\{w(t)x_1(t)\cdot w(t+\tau)x_2(t+\tau)\right\}\\
=&amp; E\left\{s(t)s(t+\tau)w(t)w(t+\tau)+\\
 s(t)n_2(t+\tau)w(t)w(t+\tau)\right\}\tag3
\end{align}$$

The step from $(1)$ to $(2)$ is intuitive, since the noise is uncorrelated to the signal we sent and received. (I'm assuming that, but it's a justifiable assumption given the problem statement.)

Now, in $(3)$ I'm tempted to say "ok, lets just pull $w(t)w(t+\tau)$ out of the expectation operator, right?", but I mustn't do that, since usually GCCs choose a $w$ based on $x_1$ and $x_2$; in other words, $w$ depends on $s$ and mustn't be treated like a deterministic function!

Information-theoretically, this means you introduce entropy into your $s(t)s(t+\tau)$ estimate by multiplying it with something that was estimated based on a measurement that includes noise. So, while $(2)$ only incorporates the noise-signal cross-correlation once, and since signal and noise are uncorrelated, with increasing length of observation, the error term in $(2)$ converges to zero. 

It's not that easy for $(3)$, especially without a specific noise model, but at the very least we can say that it should (by sheer arrogance and the intuitive application of the Cauchy-Schwarz Inequality), converge slower, if it converges at all towards the actual cross-correlation.

An oft-used practical GCC is GCC-PHAT, where $w$ is a function (usually: a reciprocal) of the Fourier Transform of an PSD estimate of $x_2$, which is pretty close to being the autocorrelation of $x_2$. Now, a reciprocal of an estimate that, should the observed duration of signal and noise be very long and if both the signal and the noise being rather whit, should contain a lot of values close to zero, is obviously rather instable, so this is an especially questionable choice in the case that $x_1$ doesn't contain noise of its own.