The common subset should probably be called something like "better C" - it won't be able to use any major (or many minor) C++ features. Basically, it will be ANSI C (give or take). Noticeably, the code for the 2nd Ed of The C Programming Language was tested using a C++ compiler, not a C compiler
The common C/C++ subset is an outdated and non-standard C though. For instance standard-conforming C++ compilers cannot compile C99 code and onward, they're stuck at a non-standard C version that would roughly be C95, minus some things that are valid C but not valid C++.
They can compile more C11 code than C99, thanks to the stuff dropped in C11.
They also support most of the C standard library additions.
As of C++20, the biggest difference is that designated initializers must follow field declaration order, restrict and _Generic aren't supported, and the type safety semantics are stronger.
There's also a handful of small differences like this, e.g. in C it's possible to construct a value and take the address for a pointer argument right in the function call:
func(&(bla_t){ .x=1, .y=2 });
This is completely valid C, but invalid C++ (C++ has references as "workaround" for such adhoc-constructed parameters which shouldn't be passed by value).
