Polyester is more difficult to dye than nylon. It requires very high heat and extreme pressure, using disperse dyes.
For polyester apparel fabrics, dyeing is usually done using a jet dyeing machine. Jet dyeing polyester also builds bulk through shrinking and helps create a soft hand.
Industrial polyester fabrics (i.e. 200, 300, 600 and 1800 denier polyester) typically use high temperature/pressure jigger dyeing. Jet dyeing these heavier polyester fabrics generally shrinks them too much, and results in a higher working loss.
In general, polyester is known to have better colorfastness than nylon.
Fun Fact: Although polyester typically has better colorfastness than nylon, nylon 6.6 bright yarn, actually exceeds polyester in colorfastness!