Buy both, or heck all three (including .17 Hornet)
I have no personal experience shooting coyotes with a .17 calibre anything.
But I have owned three or maybe four rifles in .17 Rem and two in .17 Fireball (and just got a .17 Hornet). As rpcw said, the .17 Rem gets dirty in a hurry, and groups expand a LOT. I used to clean my .17 Rem after six shots, maximum nine (shooting three shot groups) or accuracy would go downhill in a hurry. I did not have this experience with the .17 FB (and have not yet shot the .17 Hornet).
Based on shooting PAPER ONLY with the .17s, I would not advocate their use for 300 yard shots on coyotes. As others have said, the small bullets don't do very well in the wind. I admit I do not recall the ballistics of either cartridge, but the .17 Rem is a pretty flat shooter. Having said that, most folks are not particularly gifted when it comes to estimating distance, and the longer the distance the less accurate MOST people are at judging it. So a rifle sighted at 200 yards (for example) might only have a drop of three to four inches at three hundred, but at 350 that drop might be six or eight inches. These factors influence my opinion as to whether a .17 calibre rifle is a suitable choice for MOST people who wish to hunt coyotes at longer ranges.
JMOYMV
Doug