I use both. Straw mainly for the nesting boxes. During the long winter days when they are kept in I through a handful of cedar in for odor control. Some say cedar is bad but I've had no issues with it. Plus I only trhough a handful or two in at a time.
Hay is cut field grasses, full of leaves etc. It's saturates with water easily and will compact and rot quickly.
Straw is what's left over after harvesting wheat or oats. It's much much cleaner and is hollow which makes for decent insulation. It's sheds water better and will not rot as quickly.
I started buying straw... and at least one of my hens started to lay... I ran out and went back to shavings since they were cheaper.... no more eggs. I fairly certain that it is the molting and the shorter days... but, is it also possible that she preferred the straw and is now reluctant to nest in the shavings? They all liked burrowing and nestling down in the straw... but now all roost since the straw went away