When I declare a global variable in two different source files and only define it in one of the source files, I get different results compiling for C++ than for C. See the f
This isn't valid C either, strictly speaking. Says as much in
6.9 External definitions - p5
An external definition is an external declaration that is also a definition of a function (other than an inline definition) or an object. If an identifier declared with external linkage is used in an expression (other than as part of the operand of a sizeof or _Alignof operator whose result is an integer constant), somewhere in the entire program there shall be exactly one external definition for the identifier; otherwise, there shall be no more than one.
You have two definitions for an identifier with external linkage. You violate that requirement, the behavior is undefined. The program linking and working is not in opposition to that. It's not required to be diagnosed.
And it's worth noting that C++ is no different in that regard.
[basic.def.odr]/4
Every program shall contain exactly one definition of every non-inline function or variable that is odr-used in that program outside of a discarded statement; no diagnostic required. The definition can appear explicitly in the program, it can be found in the standard or a user-defined library, or (when appropriate) it is implicitly defined (see [class.ctor], [class.dtor] and [class.copy]). An inline function or variable shall be defined in every translation unit in which it is odr-used outside of a discarded statement.
Again, a "shall" requirement, and it says explicitly that no diagnostic is required. As you may have noticed, there's quite a bit more machinery that this paragraph can apply to. So the front ends for GCC and Clang probably need to work harder, and as such are able to diagnose it, despite not being required to.
The program is ill-formed either way.
As M.M pointed out in a comment, the C standard has an informative section that mentions the very extension in zwol's answer.
J.5.11 Multiple external definitions
There may be more than one external definition for the identifier of an object, with or without the explicit use of the keyword extern; if the definitions disagree, or more than one is initialized, the behavior is undefined (6.9.2).