Yes, tire pressure can be different front and back and can/should reflect weight distribution. What you want to shoot for is roughly equal tire drop while riding so that your traction characteristics are about even or slightly favor the front. If you fill the tires equally within their recommended range and then load the back more heavily, the rear contact patch will be larger than the front which will affect handling and could contribute to a dangerous front wheel slideout on a wet corner. Granted the difference wouldn't be huge but when you are pushing your luck on a wet downhill corner, every little bit helps.
The exact tire pressure you want will depend on a number of factors; your weight, riding style, load, and your preferred balance between rolling resistance and traction/comfort. Higher pressures allow a tire to roll easier at the cost of some traction and ability to absorb road vibration and small bumps. Excessively low pressures increase the chance of pinch flats.