Only if putting them away for a period of time, helps keeps the dust off them which can really build up over months of non-use. I have a few pairs of boots and shoes that see no use during the winter so I usually condition the leather and put them in bags at the start of winter. I don't put them in boxes. Boots fit like crap in the Alden boxes, I wouldn't put them back in the box for that reason alone. I've also seen the speed hooks create indents in the leather from being boxed in their crappy too-small boxes. Shoes are different and I'm sure fit fine, plus then you have the ability to stack them which is nice.
Any shoe that will likely be worn in the next couple weeks does not get put in boxes or shoe bags though, seems like way to much work for something that will be getting wear soon enough.
I bike to work so I use the shoe bags when packing the shoes into my panniers. It helps keeps dirt from the sole transferring to clothes and keeps button and zippers from scratching the leather of the shoes.
^those look like cigar shell indy boots, available at shoe mart, frans boone, and a few other retailers. what looks different is the machine stitching verse hand stitch that alden ususally does on shell. the only other machine stitched cordovan indy i've seen are mcarthur's whiskeys....