It sounds to me like he's overwhelmed by your expectations. Don't gauge your son's development by comparing him to others. Your 2 year-old cousin really is the exception. Most kids still struggle with buttons, zippers, and laces until they're about 3. And still many kids can't tie their own shoes at age 4. If he's not potty trained fully, this is really an unfair expectation, as many kids have trouble with diapers and snap on type tops.
You might want to invest in toys that teach children how to dress. Stuffed dolls with buttons, zippers, and laces provide a non-confrontational way for them to develop the fine motor skills needed to dress on their own. There are also wooden puzzles that can serve the same purpose. I think Melissa and Doug brand has them.
If you want to encourage independence, I'd recommend setting out a few easy to wear outfits for him the night before, and let him choose which to wear in the morning. Invest in elasticized pants and sweat suits, loose t-shirts and pullover tops that don't require buttoning or anything complicated, if you want him to try and dress himself. Velcro strapped shoes and slip-ons will also help with independent dressing.
You want him to gain confidence more than anything. So don't put pressure on him, and make it as easy as possible until he establishes a dressing routine on his own. Most important, don't criticize him if he has trouble putting clothes on, or looks sloppy. It's the effort that counts. With time, he'll continue to develop the skills to dress on his own and in the meantime he'll develop the confidence to get the "snags" worked out. I'd say, expect to continue helping him for a while..at least until he's 3 to 3 1/2. If he gets it sooner, be happy and think of it as a cherry on top!