For me, the way I think and write, it would take hours to do justice to your question, especially if the answer is expected to take all types and styles of pizzas into account. The best way for you to answer your own question is to take a given dough recipe and modify it 1) for short term fermentation at room temperature (called an "emergency" dough), 2) for long term fermentation at room temperature (say, 24 hours), and 3) for cold fermentation (a day or more). Then, decide for yourself what the pros and cons are.
You can see examples of emergency doughs at http://www.pizzamaking.com/forum/index.php/topic,8297.0.html
. You can see examples of long room temperature fermented doughs in the thread at http://www.pizzamaking.com/forum/index.php/topic,7225.0.html
. There are a lot of recipes on the forum for cold fermented doughs, from a day to several weeks. If you are interested in a NY style, you can find many examples in the collection set forth at http://www.pizzamaking.com/forum/index.php/topic,11860.0.html
If I were to attempt to generalize on the subject, I would say that a cold fermented dough should develop more byproducts of fermentation that contribute to the taste, color, aroma and texture of the crust of the finished pizza. However, you can achieve similar results using a long room temperature fermentation, for example, 20 hours or more. A dough that ferments at room temperature for a few hours will have fewer byproducts of fermentation and, arguably, not have as much flavor and color in the finished crust. But, taste is a matter of personal preference. There are many people who prefer "emergency" pizzas over those made from long fermented doughs.