Q:

How to map Ctrl+A and Ctrl+Shift+A differently?

In a terminal, one cannot distinguish Ctrl+A and Ctrl+Shift+A as they both emit the same key code, so I can see why Vim can't do it. But gVim, being an X application, can differentiate Ctrl+A and Ctrl+Shift+A. Is there any way to map those two things differently?
For starters, I'd like to do something like the following: Make "paste from clipboard" work like Gnome terminal, while keeping Ctrl+V to the visual mode.
:nmap <C-S-V> "+gP

A:

Gvim doesn't do it because vim cannot do it (under normal circumstances).  Sorry, but that's just how it is.

However...
Some terminals (e.g., xterm and iterm2) can be configured to send an arbitrary escape sequence for any combination of keys.
For example, add the following to .Xresources for xterm to send <Esc>[65;5u for CtrlShiftA. You can then map that in Vim to <C-S-a>.  (65 is the decimal Unicode value for shift-a and 5 is the bit for the ctrl modifier.  The u in this case stands for "unicode".)
! .Xresources
XTerm*vt100.translations: #override Ctrl ~Meta Shift <Key>a: string(0x1b) string("[65;5u")

iTerm and [u]rxvt can also be configured to do this (examples not provided).
More info: http://www.leonerd.org.uk/hacks/fixterms/

A:

As already pointed out, there are no ways to map <C-S-A> differently from <C-A>.
However, using tools like autokey (for linux & windows) or autohotkey (for windows), you can remap <C-S-A> to send a different key-stroke(s) for specific applications.
e.g. On my system, I have this setting in autokey:
$ cat ~/.config/autokey/data/gnome-terminal/ctrlshifta-gnome-terminal.py
#ctrl+shift+a sends '<S-F1>a'
keyboard.send_keys("<shift>+<f1>a") # Note that `f` in `f1` needs to be in lower case.

Assign it these properties: 

keyboard-shortcut as ctrl+shift+a
window class: gnome-terminal-server.Gnome-terminal

Then your ~/.vimrc can create mapping for <S-F1>a to do whatever you want.

Notes:

I have used <S-F1> as kind of leader key for detecting <C-S>. This was because my terminal did not accept <F13>-<F37> etc keys. If your application supports it, (gvim does I think) using those keys is recommended.
I mainly vim in gnome-terminal. So I used window class = gnome-terminal-server.Gnome-terminal as filter. Modify it to use gvim if you want. autokey supports a button for capturing any other window's properties like class/title.

A:

If what bothers you is loosing existing C-V functionality, you can use C-Q instead. See, :help CTRL-V-alternative.

