BER performance of a colored noise
Even though the variance of the noise is the same, the power spectral density of the noise has changed.  Thus the fraction of the noise power that makes it through the matched filter has changed.  If the variance remains the same but more of the noise power is concentrated outside the bandwidth of the matched filter, then the performance may improve because the noise power within the signal bandwidth has decreased.  If the variance remains the same but more of the noise power is concentrated inside the bandwidth of the matched filter, then the BER performance may degrade because the noise power within the signal bandwidth has increased.

When the noise is not white, the filter matched to the pulse shape does not maximize SNR anymore.  The Wikipedia article on matched filter addresses this situation and shows the new matched filter to be proportional to $R_v^{-1}s$, where $R_v$ is the noise covariance matrix and $s$ is the signal vector.  Using the same notation as the Wikipedia article, if you kept the filter matched to the original pulse shape $s$, your signal component would be $\alpha s^{H}s$, your noise component would be $\alpha s^{H} v$, and the SNR would be

$$\frac{(s^{H}s)^2}{s^{H}R_vs} = \frac{s^Hs}{\bar{s}^HR_v\bar{s}}=\frac{E_s}{\bar{s}^HR_v\bar{s}}$$

Here $E_s$ is the energy per symbol and $\bar{s}=\frac{s}{\sqrt{s^Hs}}$ is the signal normalized to unit energy.  This just expresses as an equation what we already concluded: that the SNR will depend on the noise correlation (or power spectral density) with respect to the signal.

If you applied a noise whitening filter, then $R_v$ returns to a scaled identity matrix but you have now introduced intersymbol interference.