I have extensively used both. I prefer the Iodine tablets (potable aqua). With the Aquamira, you have to let the solution mix for 5 minutes before adding it to your water. I found that I was managing to forget about the solution, and knock the little cup over before it was done mixing way too often to be preferable. With iodine, there is no waiting for the solution to mix at all, just fill your bottle, drop the tablets in, and go.
Additionally, Aquamira comes as a liquid, and the bottles a little bit less durable than desired. On several occasions, I opened my pack to find that one of the bottles was cracked and had leaked out all of it's contents.
Another source of aggravation with Aquamira was the two bottles. It sure is fun getting ready for a trip and only being able to find one bottle but not the other one! I eventually started duct taping the bottles together as soon as I opened the package.
The one thing that Aquamira has over iodine tablets is the taste. You don't really notice a thing. However, I think you can get taste neutralizer tablets along with the potable aqua that gets rid of the iodine taste. I've never used them, though- the taste isn't that bad, and while you may or may not develop a fondness for it, you get used to it.
It should also be noted that the long term effects of iodine on the body aren't that great. On the fine print for the iodine tablet bottles, it generally says "for emergency use only."
Additionally, Aquamira is essentially the same thing as bleach. Some (cheap) hikers just use a medicine bottle and dropper with bleach instead. I'm not sure how this compares to Aquamira in terms of how healthy (or unhealthy) it is for you.
One drawback to chemical treatments in comparison to filters is that you must wait 20 to 30 minutes before drinking the water. In colder temperatures, the treatments will take even longer to work. With a filter/pump, it takes longer to get the water, but it's ready to drink immediately.
One thing to remember when switching from a filter to a chemical treatment is the necessity of "threading" your bottle. When you fill and treat your bottle, there will be water stuck in the threads at the top that won't be treated by the chemicals. To counteract this, after filling the bottle and dropping in the treatment, close the lid almost the whole way, and turn it upside down, allowing some water to dribble through the threads. This allows the treated water to filter through the threads, killing any bacteria in them.