Why is this HST WFPC2 image so much fuzzier than the WFC3 image of the same field?
As Rob Jeffries suggested, in the WFPC2 observations the progenitor star fell located on one of the wide-field ("WF") chips of WFPC2. (I verified this by downloading one of the preview images from the Mukulski Archive.) This was purely by chance, since the WFPC2 observations were taken years before the discovery of the failed supernova. It's good luck that the region of the galaxy where the failed SN took place was imaged by WFPC2, and only slightly bad luck that it wasn't in a region imaged by the planetary-camera ("PC") chip.

WFPC2 is/was an array of four equal-sized CCD chips, each with 800 x 800 15-micron-wide pixels. However, the optics were set up so that 3 of the chips (the WF chips) were imaged using a focal ratio of F/12.9, while the fourth (the PC chip) was imaged separately with a focal ratio of F/28.3. In essence, a magnified optical beam was focused on the PC chip, giving a scale of about 3.0 milli-arcseconds per micron, or 0.0455 arc seconds per pixel. Meanwhile, the WF chips received a scale of 6.67 milli-arcseconds per microns, or 0.097 arc seconds per pixel. This undersamples the actual resolution of the telescope (which has a point-spread function with a full-width half-maximum of about 0.09 arc seconds in the I-band).

The UVIS (UV-visual) part of WFC3 has two 2k x 4k CCDs (with, I think, 15 x 15 micron pixels) and a focal ratio of F/31, which works out to 0.0395 arc seconds per pixel.

So in practice, WFC3 will always resolve things better than the WF chips of WFPC2 can, because its pixels can properly sample the telescope resolution and the WF pixels cannot.

You can see the difference in the upper "progenitor" (WFPC2) images, compared with the lower "2015" (WFC3/UVIS) images from the actual paper:



Note the stronger visible pixelation in the WFPC2 images (top two panels), due to the fact that the WF pixels are larger in angular size. 

I suspect any other apparent differences in the press-release images are probably due to the processing used to create the color image. In particular, the side-by-side (WFPC2 vs WFC3/UVIS = "before and after") color image from the press release shows both images at the same spatial scale, as part of a single image file, which means that the same pixel density is present throughout the combined image. But since each original WF pixel is about 2.5 times the (angular) size of a WFC3 pixel, the WFPC2 image had to be rebinned to match WFC3 pixel scale (1 WF pixel = 6.25 WFC3 pixels) before the combined (JPEG/PNG/TIFF) color image could be made. The interpolation used for this rebinning process may be what introduced the extra smoothing in the WFPC2 image.

[Edited to add comment about rebinning used to generate color press-release image.]