I think they both want to win. Cutler has some God-given talent that Hoyer doesn't have. I've never seen a QB in the NFL with a better combination of arm strength and also the ability to put the ball in a tight window for receivers. Hoyer couldn't do that if you put a gun to his head and threatened to shoot him. But with Jay you take the good with the not so good.
I was being a little facetious there because of course Hoyer wasn't playing to lose but he was playing to not be the guy a loss could be hung on which is just the opposite of the way Jay plays.
If you never even challenge a defense eventually they just sit on your favorite sure thing routes and take them away. Then you can't complete a 15 yard pass on 3rd an long when you need to and end up dumping the ball off making it that guys job to get to the sticks. IMHO that's more or less what Hoyer was doing and it's also what the Chicago media called him out for doing.
In Hoyer's world (and maybe Fox's too) his job was to not lose the game and it was up to someone else to win it. Good luck even winning a division title with that approach when you have to beat an Aaron Rodgers every year.
I think it comes down to different philosophies. I think Hoyer thinks that the way to win is to protect the ball first at all costs where as Cutty thinks that you need to go for it. Obviously, I prefer Jays approach. Hoyer may have the approach he has due to a combination of his abilities, maybe Foxes beliefs and maybe even a fear of the backlash Jay has received for taking risks. I think he has proven that while protecting the ball is paramount, it does not guarantee success. You must protect the ball while aggressively moving it up field.