Thank you tdollimore and to all who have posted and commented on problem and possible fixes.
I finally gave in to the annoying daily requests to update iOS and iPhone 4S will no longer connect to Wifi saying password incorrect (3G is OK)
The cause remains a mystery. It is also rather disappointing to see from this thread that after so many months and iOS 9 updates this is still an issue.
As many have reported it is also in my case most likely NOT a network/router issue, iPhone 4S has problems connecting to two separate wifi networks, yet multiple other iOS devices connect to both Wifi networks without problems (including another 4S; both old and new iPads and AppleTV's as well as MacBookPro's) .
It is also not a SSID issue. All SSID's are visible, the iPhone 4S does see and display the SSID's but can't connect (asking for password, which of course was previously not needed, and then reporting that password entered is not correct - for both routers' main networks and also one guest network - yet passwords are definitely all entered correctly.)
It is not outdated firmware as all Routers' were coincidentally updated just a few weeks ago (and as other devices have no troubles routers do not seem to be the cause of wifi issue)
Is not VPN as iPhone doesn't have VPN . . . and coincidentally an iPad with VyprVPN connects fine (with ios9.1 . . . will not risk updating to 9.2) as do a couple of OSX El Cap MacBookPro's.
It is not Wi-FI assist as this has been disabled for iPhone 4S and so not an available setting.
Tried (in this order with no success):
Forced restart iPhone
Resetting iPhone network settings
Rebooting Routers
Full Backup up of iPhone manually to 'This computer' before resetting and restoring from local backup
(this final step essentially a disaster as restoring from backup *failed to find* 124 apps, second attempt to restore returned sync error, third attempt reset and restore again returned lost apps error . . so iPhone essentially wiped of most Apps and still no wifi connection! Now manually reinstalling key apps one by one via App Store over 3G :-(
Didn't try: Entering Google DNS (as without wifi connection there are no numbers under DHCP to change, plus iPad that is connected without problem has DNS settings (not Google) from Router, and connects fine);
Odd behaviour, possibly linked? (or just coincidental):
For some weeks getting error message "This iPhone hasn't been backed up in [large number] weeks . Backups happen when this iPhone is plugged in, locked, and connected to Wi-Fi" despite all conditions being met (until iOS update and loss of wifi connection) *large* number now at 143 weeks (?)
Asked to trust computer when connecting, this 'trust' not being remembered so being asked every time (iPhone and computer end).