A rational person might assume that the program from which you export your media wouldn't have a noticeable impact on the quality of the final image, especially if the export settings are identical in both programs. A recent test by filmmaker Noam Kroll might just teach us to think twice before making assumptions.
First, a little bit of background on Kroll's test. Having noticed that exporting from Adobe Media Encoder yielded quicker results than using the same settings and exporting from FCPX, he tended to use Media Encoder for the bulk of his exporting. When a recently exported project came out with some nasty compression artifacts, blocky rendering of certain areas, and a noticeable change in color quality, Kroll put on his detective's hat and tried exporting again from FCPX. To his, and soon to be your, surprise, the exported result from FCPX yielded significantly higher image quality with the EXACT same export and compression settings.
Don't believe it? Have a look for yourself. According to Kroll, "both FCP X and Premiere Pro were set to output a high quality H.264 file at 10,000 kbps." The image on top was exported from FCPX and the bottom was exported from Premiere Pro.
In the shots above, you'll notice more blocky compression artifacts in the version exported from Premiere, especially on the lower part of the woman's face, and there's a fairly significant reddish hue that's been introduced into the midtones and shadows of the Premiere export. Here's a version of the same shot that is cropped in on the woman's face by 400%. This is where the difference between the two starts to become painfully obvious. Again, FCPX is on top, and Premiere on the bottom.
Here's the conclusion that Kroll came to in his post.
After seeing this I can confidently say that I will not be compressing to H.264 using Premiere Pro or Adobe Media Encoder any more. [sic] The image from Premiere is so much blockier, less detailed, and muddy looking, not to mention that the colors aren’t at all accurate. In fact I even did another output test later on with Premiere Pro set to 20,000 kbps and FCP X only set to 10,000 kbps and still the FCP X image was noticeably higher quality, so clearly something is up.
It's really difficult to speculate as to what's going on behind the scenes that's causing such a drastic difference in results between the two programs. However, what is clear is that you should take caution when exporting to h.264 from Premiere and Media Encoder. Regardless of the program that you're using, perform your own tests and make sure that the export process is leaving your media with a visual quality appropriate for the delivery medium.
The good news here is that Adobe is extremely receptive to feedback from their user base, and their Creative Cloud subscription model allows them to roll out updates with a much higher frequency than they could with the boxed version of the Creative Suite. If more people are experiencing these problems and reporting it to Adobe, chances are that we'll see an update with fixes sometime in the near future. With that said, I have no idea how Adobe handles the technical process of exporting, so it could very well take a complete overhaul of how the program encodes h.264 to fix the problem.