testing your internet connection with a browser isn't really the best test. A browser means world wide web and http, which is a protocol/service that runs on the internet.
Connected to the network just means you have IP configured (IP address, subnet mask, default gateway) which is just your LAN or WLAN connection.
The Internet access bit probably just pings an MS server somewhere, and may not even use DNS to do so.
Webpages use the DNS server to convert the URL to an IP address. So, you might just have an issue with DNS servers.
In theory, theory and practice are identical.....in practice they aren't!