I have been meaning to document something on this subject for a long time. Meanwhile I hope this helps.
Firstly, I will explain the numbers and perhaps dispel some myths. The 1st digit describes the chain pitch between pins in 1/8 inch increments. The 2nd and 3rd digits are the internal chain width as a decimal multiplier of 1/8 inch.
Eg.. 630 chain is 6 x 1/8 pitch and 3.0 x 1/8 wide..... 530 chain is 5 x 1/8 pitch and 3.0 x 1/8 wide.
Or to put it another way... A 530 chain is 20% shorter pitch requiring a 20% longer chain run than a 630 chain. The internal width between the plates is exactly the same. Weight and external width saving is due to thinner plates... about 0.2mm each side giving a 720 gram weight saving per 100 links
The attached from DID is a good guide to comparing.
As said the biggest advantage is a bigger choice of chains and one of the things often overlooked is the significantly longer life of X ring chains over O ring type, at least 50% greater wear resistance up to 100% for premium chains.
The only minor downside of 530 chain is equivalent sprockets are not always available off the shelf.