I assumed OP is not flying paid F & C tickets since he/she is interested in *G benefits. I am not sure what *G benefits a paid F or C ticket doesn't provide.
LAX
Depending on place/program, but off the top of my head:
- *G checkin, where available (if in C, *G may give access to F checkin and/or separate *G checkin )
- access to Scandinavian lounges in CPH/ARN, even with a C ticket -- I assume that it may be similar elsewhere too?
- probably some priority-waitlist-thing
- in general, "more miles"
I am sure that there may be some lounge-thing-in-the-US-when-traveling-domestic, but I can't figure it out

I think, though, that a non-US *G is recommended since US *G means no lounges within the US when flying domestic.....