It varies across birds, as, for example, a cockatoo has such different requirements from a canary. A cage that is too large for a small bird can cause a hazard to the bird getting it's head caught between the bars.
From what I've seen the cages for bigger birds with the bigger spacing often have stronger wire to make up the bars, as the cage will, usually, be bigger and needs to be stronger.
Bird Cage Bar Spacing Guide
For metric users:
Small birds 1.3 - 1.6 cm
Medium birds 1.3 - 1.9 cm
Large birds 1.9 - 2.5 cm
(1) It's better to use a smaller bar spacing than a larger one in the range given for a species. A smaller size makes it less likely that a bird will poke its head through the bars and get stuck.
(2) Beware of the very small finches! Small waxbills (e.g., orange-cheeked), tiny manikins (e.g., bronze-winged) and other tiny finches can squeeze through 1/2 inch bar spacing. Use 3/8 inch for them.
Thanks to comment by Tom Gaskill
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