From the documentation:
ACCESS_NETWORK_STATE:
Allows applications to access information about networks
INTERNET:
Allows applications to open network sockets.
In short, the INTERNET permission lets you use the internet, whereas ACCESS_NETWORK_STATE will just give you information about the network, such as whether you are connected to a network at all.
You can use either one without needing the other. If you don’t check that you have a valid network connection before trying to use the Internet, your HTTP requests will simply fail.