Does not make much different either way. All I can think of is...
Some claim that the sun's UV light will damage the spare tyre's rubber after many summers, but the 4 main tyres in use are exposed to the sun anyway. Although the 4 main ones are mostly covered by wheel arches and the ground, leaving the side exposed, the spare tyre on the back door is mainly totally exposed.
Sometimes it may help protect your tyre from people, someone letting the air out or slash it with a knife, but it is really pointless considering that they could still slash it anyway, by slashing through the cover. Unless you use one of the hard cover types, but it's bit expensive to buy.
Some may like to personalise their cars by using a personalise cover. You could use some graphics or text on the cover, it's only useful if you find it boring looking at the spare tyre every day, or if you're self employment and want to have your company name on the cover.
When I got mine second hand, it did not come with a spare tyre cover, either the previous owner didn't want it or lost it, I have no idea. But at moment, I'm thinking of having a go buying a cover, I need to put one on and see how I feel, so I could compare covered against uncovered, but unless I experience it for myself, all I can say is that it hardly make much difference.
