what does control t do different?

I can modify a click using the option, command, shift, and sometimes a combination of these modifiers. 
So what does control-T do that these modifiers don't? 

By @Mr. Content

When you say “I can modify a click,” if you really mean distort a selection, those modifier keys work only when the transform controls are visible around the selection. If the transform controls are not visible, you cannot use those keys to distort a selection.

There is more than one way to make those controls visible, and this is the answer to your question. They are simply different ways to make the same transform controls visible.

One way is to select Show Transform Controls in the options bar, when the Move tool is active. This is the first part you see in the demo below. If you select Show Transform Controls, you can now use those modifier keys as you drag any handle. What makes this method different is that when Show Transform Controls is active, the controls are always visible when you make a selection. Some people like that, some people don’t. The ones that don’t want to see the handles all the time use the next method.

Another way is to choose Edit > Free Transform, the command with the Control-T (Windows) / Command-T (macOS) keyboard shortcut. This is the second part you see in the demo below. With this method, the transform controls are hidden until you use this command or its shortcut to see them.

So, if you are working with Show Transform Controls on, that might be why you don’t need to use Free Transform (Control-T).