This.
I have the same issues, and I'd guess everyone who uses FRFR on stage and goes direct to FOH has this issue. I use 2 CLR's on stage, stacked behind me like a vertical 2x12. (I used to use them as wedge monitors, but I like them behind me much better.) We own our own PA that we use at some clubs or private gigs, and it's very nice (QSC KW153's over KW181 subs.) My Axe sounds NOTHING like my CLR's through the QSC mains. It's dull and lifeless, with all of the top end gone. We have to apply an EQ curve at the board to compensate. (The board is a digital A&H, and has a nice parametric on it.)
If you wonder why I don't do it on the Axe, it's because the CLR's and the FOH both come from OUT 1 via a line mixer. This is because I use Out 2 for something else (I send a signal out to my pedal board for my talkbox, essentially using the Axe as a switcher.)
Anyway... this is just the difference in speakers. CLR's don't have a lot of low end in them. The highs are very clear and pronounced. Supposedly they are very "flat" and accurate. I don't know, I'm not an engineer... all I know is they sound great and get really loud.
But... for me.... all that "accuracy" doesn't do me a lick of good for dialing in tones, because that is not what the FOH puts out, so there is a big difference. I have considered changing my backline setup to more closely match our PA so when I dial in tones, I get the same at FOH. Of course, that all goes out the window when we play venues with house systems though.