It's easy to understand why there's so much confusion surrounding the differences between earbud and in-ear headphones. The two designs are sometimes referred to interchangeably, but they are two very different types of headphones. Earbuds rest on the outer concha ridge of the ear, located in the center of your outer ear. In-ear or ear-canal headphones are placed inside the ear canal, sealing the listener off from environmental noise.
A few in-ears -- like the Etymotic ER-4 -- fit deep into the ear canal to provide maximum isolation. Most in-ears are more comfortable "shallow fit" designs that provide less but still substantial noise reduction.
One not-so-obvious advantage of blocking outside noise is that since you don't have to play music loud enough to overcome the din, you can listen at much lower, safer levels. In-ears' noise isolation is a purely passive function; it doesn't require batteries or electronics. In-ears' isolation is good, but not as effective at hushing the din as the better noise-canceling in-ear or full-size headphones, which always use batteries and electronics. I find that noise-isolating headphones tend to sound better than noise-canceling headphones.
Some people find the process of inserting in-ear headphones uncomfortable and don't like being cut off from the environment, so they will probably find earbuds' less-intrusive fit more comfortable. Earbuds also have another potentially significant advantage over in-ears: They let you hear more sound in your environment. With earbuds, you won't feel sealed off from your surroundings, and that's definitely a plus if you're exercising, riding a bike, or walking on busy streets. I'm referring to the sort of earbuds that come with phones and iPods.
For best sound quality, I prefer in-ear headphones. Finding great-sounding earbuds is next to impossible, but Phiaton's PS210 "half in-ear" headphone ($159) bridges the gap; it's literally a hybrid of the two types of designs.