Why is a squared cosine window function commonly used for 2D spectra?
The difference in window function is a result of the parameter being optimized for the respective experiment.  Exponential multiplication of the Free-Induction Decay (FID) weights the beginning of the FID where the amplitude is at it's greatest.  Weighting the beginning of the FID results increases the signal-to-noise ratio, since the noise should have the least impact with the strong signal.

For 2D NMR, we are usually more interested in resolution to aid in differentiating the various peaks and cross-peaks.  We can always take more scans to try and increase the signal strength, so resolution enhancement becomes the key objective.  A pure sine function offers increased resolution at the loss of signal enhancement.  The shifted sine function (or pure cosine function with SSB=2), provides a good mixture of the two objectives.  The enhancement doesn't weight the early parts of the FID as heavily as exponential weighting, but more than a sine function.

The use of SSB=2 for a cosine function is used most often in 2D experiments where we need a usually more significant amount of signal enhancement, such as TOCSY, NOESY etc.  sine enhancement is used most often for COSY, to reduce filtering out peaks from overlap.

--UPDATE--

To answer some further questions posted in the comments.  Using a gaussian weighting for 2D-NMR data is not as preferable as cosine or sine weighting.  The gaussian weighting has a much more narrow distribution, and rapidly approaches zero.  This can prematurely truncate the FID (by essentially weighting the later portion as zero), which introduces artificats into the resulting spectrum known as "truncation wiggles".

Squaring the cosine (which remember is just the sine function shifted) is done both to slow the increase and decrease of the weighting function over the course of the FID.  At the beginning of the FID, this allows for a greater amount of sensitivity to the early parts, while sacrificing a small amount of resolution.  Additionally, the slow decay also reduces the truncation artifacts discussed in the first question of the update.