Not trying to be combative or difficult here.. but..
How would you know ??
By that I mean.. different speakers (PA vs bass cab), room reflections, dynamics, back wall bounce, listening position (on stage vs out front), etc.
Out front, you are probably hearing stage wash combined w/ PA, room acoustics. It won't sound like what's coming out of your amp - even tho it's the SAME signal source!
I know for a fact (and from experience) that a clean, unaffected, flat EQ'd sound you send to FOH and hear on stage (with flat EQ on FOH console) CAN [and very often] sounds different thru the PA, both FROM the stage and out front. I'm pretty sure that @Wasnex and
@4Mal will confirm that!