Infants born more than three weeks before their due date (in other words, before 37 weeks gestation) are considered premature. We don't expect such infants to be as advanced in their development as other infants of the same age, because they didn't have the chance to mature fully in utero before they were born. That is why we "correct for prematurity" on most tests of infant development. If you might expect a full-term infant to do something at 20 weeks of age, for example, you might not expect an infant born 10 weeks premature to reach the same milestone until closer to 30 weeks of age (20 weeks of age + 10 weeks correction for prematurity). This method of correction for prematurity is really just an estimate. Infants that are ill after they are born may take longer to catch up (since they don't learn well while they are sick), and some infants will need less time to catch up (if infants are healthy, they can learn some skills faster once they are born than if they had remained inside the uterus, since there are different opportunities for them to learn and interact with their environment once they are born).
We only correct for prematurity up until 2 years of age on most tests because after that point, a number of weeks doesn't make that much of a difference. It's kind of like the difference in ages of spouses. When they both are young, it seems like a big difference if one spouse is 19 and the other is 24, but by the time of the 25th wedding anniversary, the difference doesn't seem that great -- there is often a real difference in maturity between a 19- and 24-year-old, but not that much of a difference between a 44- and 49-year-old.
In many areas of health and development, such as getting immunizations, premature infants are treated based on their actual age. So your infant should be getting his first round of immunizations in a few weeks, at his 2-month checkup, unless there are major medical issues that preclude this. As for how you should treat your own child, I would suggest you treat your premature infant just as any parent should treat any infant -- based on what your child's current needs and strengths are. Once a premature infant has reached his due date and is well enough to be discharged from the hospital, you probably no longer need to treat him "as a preemie." If your child has some remaining medical problems that were due to being born premature, then you obviously need to attend to those needs. Ask your health care provider if you have specific concerns about your son's health. But if he is otherwise healthy, it's time to view him as a healthy child. He may have been born earlier than expected, but he's here now and that's what matters most!