Switching from a quill stem to an stem adapter?
Quill stems are in general better for adjustment -- you have far more height adjustment than using a quill to threadless adapter like this:



You'll probably end up combining the threadless adapter with an adjustable stem like this:



in order to tweak the adjustment.

They do make adjustable quill stems:



which will give you more adjustability for height and position.

The weight of your stem is negligible compared to the rest of the bike (and you), so the weight savings are inconsequential unless you're some top end racer (in which case, you probably want a new bike). So its pointless to do this for weight savings. 

Things you need to look for in either option (threadless adapter + adjustable threadless stem or adjustable quill stem) are the clamp diameter (where the bar is gripped by the stem) and the diameter of the adapter or quill stem (i.e. the part which goes into the bike). If the diameter of the adapter or quill is different, the stem or adapter obviously won't fit in the bike and won't work. If the clamp diameter is different, you'll need to buy new handlebars. 

Note that a stem only provides a bit of adjustment -- you may find that neither of them can make the bike fit for you in which case you need a different bike. I'd probably opt for the adjustable quill, since you can adjust the height more and in net it will probably be a bit cheaper if not about the same as the threadless adapter + adjustable threadless stem. 

EDIT: In the comments, Fred the Magic Wonder Dog has pointed out Soma makes a super long threadless adapter (nearly a foot in length!). This would give you a good deal of height adjustment to which you can use an adjustable threadless stem. All in all though, I think both options will end up costing about the same.