I consider myself a run of the mill rec rider, and here are my truthful experiences with a threaded headset, featured on my '05 Bianchi Brava, the last of the truly 'classic' Bravas, if that word can be applied to a Taiwan-manufactured bike (the '06 switched to threadless for good).

The threaded headset required constant adjustment. After a crash in the middle of a ride it was knocked loose, and not knowing enough or having the proper tools to tighten it correctly I was forced to ride it loose to get back home. The threading got damaged, and pretty soon the headset was coming loose pretty much on a weekly basis, even after taking it down to the shop to get it tightened multiple times. My headset and quill stem also creaked - a lot. Every hard stroke up a hill would give me a very confidence-inspiring squeak or groan from the whole setup, even with all the bolts tightened down.

I finally received a shiny new silver headset from Bianchi under warranty, and boy did it look nice. The problems went away, but even so I was turned off of threaded headsets for good. During the three months in which I owned the Brava, there was not a one to two week period when a new headset problem arose. I gotta honestly say that I'm a lot happier with my new Veloce with threadless.

Here is the small list of advantages and disadvantages that I compiled through my experience only. Please note that most people with threaded headsets haven't experienced the problems I have and like them very much, but I'm just recounting based on my story.

Threaded
-Comes loose, requires constant adjustment
-Requires a flat 32 mm wrench to properly maintain/tighten
-Creaky (for me at least)
-Lack of stiffness
-Always a little bit of play between fork and steerer, even when tightened down

-Admittedly, looks very beautiful when done right
-Adjustable

Threadless
-Non-adjustable
-Lack of 'classic look'

-Much stiffer, tighter
-Adjustment requires minimal tools

Need to add that the stems become cheaper to manufacture as well. I actually stopped at Far Horizon's answer (first responder) because it was the all that needed to be said.

Been riding for 50 (oops 51) years and have yet to have the "...comes loose..." problem on a good quality Threaded headset. Using Campy, Stronglight, or Tange (campy copy) only. Creaking stem is easily fixed with the application of a small strip of aluminum shim made from a RED BULL can. Won't even see it...Sorry old school thinking.
I have a number of bikes threadless as well as my older steel threaded ones. Still ride about 5K miles per year distributed between all my bikes, so none of my bikes are what one would deem as "wall art."
Good luck

You will need a new fork, a threadless headset and a stem. Performance sells a bundled upgrade kit with all three that seems like an ok deal to me. If you are doing the labor yourself, there is a lot more you need (you can improvise some of it).

You need to get the old headset cups out of the frame (be careful, but a flat head screw driver, a mallet and patience will do it. Then you press the new cups into the frame (you can make a home made press with a threaded rod, some washers or bushings and a few bolts). The last tricky bit is pressing the new race onto the fork. All of this isn't all that hard, but if you don't feel comforatble with all of that, have a shop do it.