Back when I researched headphones many years ago, sound quality was supposedly better with closed headphones, but I guess it depends on your definition of "better" and on what you want. I still have some old Sennheiser HD590. The cord and ear pads have both been replaced two times over its life time. They are open, which was important to me so I would still hear the telephone ring while gaming.
The pros of one type is the con of the other (personal subjective opinion):
Closed:
+ stronger bass
+ can hear more in noisy environments
+ good if you don't want to disturb nearby people in quiet environments
- bad if you want to hear what's going on "outside"
- need to vent your ears regularly due to lack of air flow
- sound stage appears very close to your head
Open:
+ can be worn for longer periods of time
+ wider experienced sound stage
+ can hear what's going on near you
- can possibly be heard from the outside, which may disturb other people
- bass not as full
- not as good in noisy environments (need to increase volume)
Other things I remember:
- good quality headphones
may require headphone amplifiers, if they have high impedance (Ω). There are sounds cards that offer this built into the headphone jack though, otherwise they'll sound really quiet and won't live up to their potential.
- good headphones
may need to be broken in before they sound like they should (? run some diverse music or noise on it for a night or two or more? *shrugs*)
Last, but not least, I also found it important to me that the headphones enclose the entire ear and are very comfy. Wearing my pair feels like gently pressing cushions to your ears and the ventilation is good enough so that I can wear them for hours without detriment.
Ah, here's a nice short video explaining the main differences between closed and open.