Neural network is a classification tool used to perform some AI algorithms (relationships, detection, prediction ...).

Social networks: a group of networks (websites, mobile applications...) for people interaction (talk, posts, videos, photos).

Thanks.

Neural Networks generally inspired by neural systems in human bodies, whereas social networks are any kind of networks that has special connections related to human relationships and activities like the network of researchers, citations, facebook, twitter, ...etc.

And of course these two types of networks has a totally different underlying assumptions and applications