Solution for WIFI connection with no Internet connectivity on S3
Having tried just about every solution proposed in this and other threads and forums to no avail, I just found something that works - changing the IP address from DHCP to static.
Before starting the steps below, log onto your router's home page, and go to the page which lists the DNS server's IP addresses. There will be two; primary and secondary. Have this info ready, then start the steps below.
1: Go into settings>>> WIFI on your S3
2: Select the particular WIFI network you are connected to by holding your finger on it.
3: Menu pops up; select "Modify Network Config"
4: Go to IP settings, select "Static"
5: A number of fields pop up. You only need to change 2; the last two which are DNS1 and DNS2.
6: Take the DNS IP addresses from your router page opened earlier and type them into these two fields on your S3.
7: Select save at the bottom of the screen
Open your browser and try accessing a web site.....
Hope this works for others... but I haven't yet fully researched the implications of doing this on other situations, for example logging onto a different WIFI network...