The network model if it means anything is letting you add pointers to the mix and changing how you reason about your data from sets to graphs. This makes it harder to declare constraints, check integrity, update sets of data, access your data in different ways, and reason about your queries. An imperative script is in no way as safe a program as a declarative query.
The most clear benefit in my view to the network model it lets you use your object code fairly seamlessly. Ok.
The problem is that this is a good trade off for programs where you don't care primarily about your data, but about your code. I'd argue that isn't true for the majority of programs that have databases at all. The issues that kill your system years down the line and give you nightmares are not code issues, they are data issues. And the code you write to fix those problems... will end up re-implementing all those annoying "heavy" bits of RDBMSs that coders seems to hate.