Published world:
*Built in adventure hooks
*Premade NPCs
*Drag and Drop modules into it with little to no adjustments
*You don't need to tell players "Here's link to my webpage with the 300 pages of history and house rules so you can get a feel for what the world is like and what characters fit into it." Players already know what characters fit into FR, Greyhawk, Dragonlance, etc.
*Most published worlds also support all or most of the published character concepts so a player who didn't read your 300 page description of your world can bring his monk/uncultured Gruumsh worshipping half-orc barbarian/Knight of the Chalice wannabe and you don't have to say "that PC won't work in this world."
*No need to think up linguistically coherent names. (Why does this city have a chinese sounding name in the middle of an Italian Renaissance style setting?) Hopefully someone already did this for you.
DIY world
*The flavor you want. It will support exactly the options you want it to support. Consequently it can be much more unique.
*Players don't "correct" you on the way the world works.
*More opportunity to explore unique cosmologies and ideas.
*A chance to create a world (it can be a lot of fun).
*You can still used published modules and supplements. You just need to be more careful about which ones you use (Great Modron March won't work if your world doesn't have Modrons and only really has 4 inhabited planes (Astral/s0pirit world, Ethereal, Prime Material and Celestia/Heaven)) change the names.