Yes, it would sound (slightly) different. The crossover will take out everything under (usually) 125Hz, sometimes as low as 75Hz, from the signal going to the full range speaker, freeing it up to spend more energy in the higher frequencies. You could opt for mono out, in which case you’d go mixer -> sub -> full range 1 -> full range 2.
Personally I’d just keep it stereo and have the sub on the side with the full range that is hooked up to it. That way the full range away from the sub will take up some of the bass to even out the spread. But with low and sub low it isn’t all that exciting and nobody will notice I am guessing.
The alternative would be to get a separate active crossover but that is money, extra gear, extra cabling and setting things up correctly. I wouldn’t bother.