I'm trying to install new apps on iPhone 4S via iTunes 12. Music, info, photos are no problem, but iTunes will not allow me to install apps I have downloaded on my Mac via the App Store. All new apps have their respective "Install" button 'grayed out'. I have had no problems up to this point installing new apps. I have changed nothing on either the Mac or the device. All of the apps I want to install are free. Over 2 GB of free space in device.
Without knowing the actual apps you're trying to install and what version of iOS you're running on your iPhone 4S, any answer you get is likely to be at best an educated guess.
There are a number of possibilities for your situation, but my first thought is that although you say nothing has changed on your Mac or iPhone, that doesn't mean something else hasn't changed.
Specifically, what I'm thinking is that many software developers are busy updating apps at present due to a number of updates (e.g. iOS 10, Xcode 8.1, Swift 3, etc). In doing so, many apps may become incompatible with older hardware and/or operating systems.
In your case you've downloaded the apps on your Mac, so there's no built in check going on as to compatibility. What I mean by this is that browsing/downloading the apps via the App Store on your iPhone will only allow you to download versions of apps that will install on your device, but there's no way for this to happen via the Mac as you could be downloading them for any number of devices.
Another factor is that your model of iPhone cannot run a version of iOS later than 9.3.5, so any apps requiring a more recent version will not install (and that's assuming you're even running iOS 9 (many iPhone 4S owners aren't).
You'd be best to use your iPhone to download apps as many developers (me included) do allow older versions of their apps to remain on the App Store for superseded devices etc.
Hope this helps!