Why is the bathtub hot water brown but not the sinks?

If it's in the tube water then it's in the sink water. Easier to see in the tub because it's deeper. You're seeing minerals (usually iron particles) suspended in the water.

You have brown hot water because you fail to flush the iron oxide( rust) out of your heater. Let me show you how,
For long life and fewer troubles you should keep your heater clear of mineral build-up by FLUSHING NOT DRAINING on a regular schedule.

Attach a hose to the boiler drain at the bottom of the tank. With the pressure on, open the boiler drain and let it run until the water runs clear. You will see a spurt of red,(rust) followed by white or yellow grains,(lime or calcium carbonate). This shouldn't take more then a few minutes.

Do this monthly to keep it clear. Now flush out your hot water lines on ALL fixtures that are affected . Now pull each aerator and clean the screens. Be sure you put them back together the same way you took them out. Don't forget to flush it out every month. Your heater will thank you for it. Hope this helps and thank you for rating my answer,

Tom