The Red Wings logo is very flat. There is one color and its traditional as with any Original Six team. I think it's okay but also that there's still room for improvement. But I'll leave it there. As for the Blue Jackets logos, it's a huge improvement over the previous logo (anybody remember the electric green bee?). At the very least, it makes better use of color than the Wings logo does.
A Blue Jacket may carry a bayonet, but chances of him shooting a bird with Red Wings are slim.
Both teams have colors in their names and yes, those colors are prominent in each logo. In fact, red is the only color in the Red Wings logo. The Blue Jackets logo prominently features the Ohio flag wrapped around a silver star in the form of a "C" as has been mentioned. A reader also pointed out that the star signifies Columbus as the capital city. And, yes, it's that cool.