Apparently Illustrator uses different means of detecting a graphics card than photoshop does. I had the same issue and it was related to WMI being broken on a windows machine.
Try running msinfo32.exe and if it comes back with an error (doesn't display computer information) it's probably the same issue.
To fix it run create the following script by saving all the text below the line in notepad. Save it as fix.bat then save it in C:\windows\system32\wbem. Open command prompt as administrator then type cd C:\windows\system32\wbem and hit enter. Now type fix.bat and let the script run. It might take a few minutes to register all the components. Once it's done restart the computer and it should work now.
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@echo off
sc config winmgmt start= disabled
net stop winmgmt /y
%systemdrive%
cd %windir%\system32\wbem
for /f %%s in ('dir /b *.dll') do regsvr32 /s %%s
wmiprvse /regserver
winmgmt /regserver
sc config winmgmt start= auto
net start winmgmt
for /f %%s in ('dir /s /b *.mof *.mfl') do mofcomp %%s