Tow haul mode can be much more sophisticated than just "shifts later".
My 99 tahoe had no tow haul mode or button. I changed the PCM to an LS1B computer and gained the ability to add that tow haul mode with a button that momentarily grounds a wire going to the PCM.
After opening the tune file in EFI Live tuning software I was able to adjust Trans shift points for speed, load, throttle position, rpm, coolant temps, etc. You can raise the trans fluid line pressures, change when the torque converter locks up and unlocks under different loads, rpms, speeds and throttle positions to limit constant hunting of gears, etc. All this is to maximize hp/tq and limit constant shifting and torque converter unlocking which keeps the trans temps cooler.
This is 2001 PCM.
I can imagine that they have the tune files and shifting strategies pretty well figured out by 2015.
Use the tow haul button!