Both have benefits, in my opinion. Obviously, most chairs that you rent on the beach are going to be more comfortable than typical folding beach chairs. And many places where you rent these chairs offer a free drink or restroom access with your rental. But carrying your own has benefits, too - you can go pretty much anywhere you want any time you want - I've noticed some places don't set up chairs until 9:30 or 10 and it's nice to get an earlier start. Also, the vendors on the beaches (bags, necklaces, massages, etc) will spend less attention to the "bring your own chair" people (I guess because we look less like tourists) - of course, if you want to buy something, you can flag them down but sometimes it is nice to sit and just read and not have to say "no thank you" a dozen times.

If you spend long hours on the beach, you may be better off renting from the various vendors/restaurants/bars on the beach. If you like to go to multiple beaches in a day, the foldable, pack able may be your best option.

Not a lot of help, am I?

Regardless, have fun!