DNS gives more control and precision to CDN provider than BGP. BGP might be good enough if you got few POP around the world but with the scale/distribution of Akamai, DNS is better suited.
One is an actual routing system that tells routers where to send data, the other is a name translation system with multiple layers of caches outside of your control.
DNS is a layer above as BGP is still used to actually navigate to the listed IP, and any large CDN will own that IP space and announce their own routes anyway.
Yeah, I'm not sure what parent is on about, BGP and DNS are not alternatives to each other, the internet relies on both of them but at different layers. Without BGP packets wouldn't know how to be routed and without DNS they wouldn't know where to be routing to, they are complementary.
Answer: Nothing. They are just IPs.
Using a combination of Anycast and DNS is going to give the best control over steering http traffic. Particularly if you own a few prefixes and can do clever addressing tricks.
With DNS and dynamic responses you are directing request to specific DC, even server, almost on every request. It may be dedicated for this traffic type (live stream different than static images etc). Your DNS server can take the hostname ("www.google.com") into consideration - BGP doesn't even know the hostname in the URL. If you wanted to do it with BGP you would need to place specific content to a dedicated /24 subnet and that is impossible considering how many IPv4 addresses are available.
BGP doesn't even consider network latency, current network load. CDN knows load on their machines, on their network link, where given content is placed. The bottleneck may be storage, network or CPU processing, different for different sites and content type. They need to direct traffic on request basis considering this and at least the hostname from the URL. That's why DNS is used first.