The Battlefield shot was originally 1080p. So that evens it out. The extra blurring is part of the game.
It's not that much farther away.
No, it doesn't! If you resize an image and then save it with low quality lossy compression it doesn't matter what size it was originally, the quality is still lost. I'm not talking about blur, I'm talking about the jpeg artifacts you can see on absolutely everything. I mean, it's hardly surprising that if you take decent footage, take a picture when you're right next to a wall that's blocking almost all of your view, compress it poorly, and save it in the middle of heavy motion blur (Because while blur looks fantastic in motion, it does so because it removes the emphasis on per-frame detail under motion), it doesn't look great.
The comparison is very poor, you're looking at two different kinds of scene without giving any respect to the technologies at play. The same way it's hard to get a decent image from a movie or TV show due to blurring, it's hard to get a decent shot under heavy motion blur - that you then ruined the image further with terrible compression... yeah.