The Dremel chuck lets you quickly & easily change accessories with shank size between 0.4mm & 3.4mm, without having to find the appropriate collet size, undo the nut, insert the collet & do up the nut again. Just twist, insert the bit, tighten, and you're done.
I've often wondered why Dremels aren't sold with a chuck as standard, as it's so much easier to use & more versatile than the collet system. Apart from the obvious answer, I think there are some advantages to the collet system for dusty applications such as routing & grinding, because this system can be easily taken apart & cleaned. I also think collets are a little more stable for some uses- for example, full size routers use collets rather than chucks.
The main disadvantage of a chuck over a collet though is that it inevitably gets filled up with dust and swarf, but unfortunately can't be taken apart for cleaning. So, the chuck gets increasingly clogged up over time. This could be remedied either by making it possible to dismantle the chuck for cleaning, or maybe some kind of dustproof rubber cover could be fitted.
The other minor snag is that the threadded nose cone of the Dremel can't be removed with the chuck in place, as the hole in the nosecone is too narrow to slide over the chuck. So, if you need to fit an accessory which screws into the threadded interface, you need to remove the chuck first before unscrewing the nosecone. This could be easily sorted by making the hole in the nosecone a little wider.