@Zolf
You must put a DNS to your Zentyal first. if your internet type connection is DHCP.
i think it is automatically add to your DNS, if your using static add you ISP DNS or
you can use OpenDNS 208.67.222.222 and 208.67.220.220.
If you already has a DNS . Use the CLI of zentyal server type
nslookup google.com you must get a resolution if not your are not yet connected to the internet or
there is somthing wrong with your internet connection.
If you already have a working internet in your zentyal gateway. you can ping google.com.
then next thing is to have your client setup.
Ok let get back to your question.
the difference between the two is the DNS for the network is for your Zentyal to go out and resolve something to the internet if you dont have this you cannot browse or ping any websiteusign their name ex.www.yahoo.com,google.com etc.
The DNS in infrastructure is a service in your lan which will give you to add domain to your Private lan. for example you created a example.local this will be your domain for your private lan. the rest will be Computer1.example.local etc so you have a naming resolve within your lan.
Sorry my english is not that good hehhe.