How big of a difference does it make to go from a 14" x 4" register to a 14" x 6"? Is there a calculation to know what sizes to put in? Or does it usually go off the type of room?

Supplu or return register... and are you changing the duct work too? Going 2 inches won't make a huge difference if any.

I changed a 7 inch flex with 4.2 R to a 8 inch with 6 R and split off to two 6 inch lines with 6 x 14 registers