All have positive and negative qualities.
Chemicals are light, but sometimes have bad tastes, and you might have to wait several hours to kill protozoans in cloudy or cold water. They generally kill viruses and bacteria quickly. (The MSR MIOX falls into this category- I personally think it is a useless device. It has all the disadvantages of other chemical treatments, but weighs more and requires batteries. It MIGHT be useful for treating very large volumes of water, though, like for a large group.) Iodine is outdated, by the way- it turns out that it may not have much activity against cryptosporidium. The newer chlorine dioxide systems are better (AquaMira is popular, and they make tablets which are quite convenient). You can combine chemicals with a very basic filter- like the Frontier Pro- to remove the protozoans quickly instead of waiting 1 to 4 hours.
Filters are fast and convenient, but the vast majority don't stop enough viruses, which is why many companies package them with little dropper bottles of virucide. The ones that have filters fine enough for viruses are generally slow. You have to be careful not to let a filter freeze or they may crack (especially ceramic ones), and they may occasionally need to be cleaned.
(But I can't recommend Sawyer filters enough- they make a 0.02 micron microtubule filter rated against viruses, and it is the one they recommend for third-world travelers. Even their "lesser" 0.1 micron 4-way filter probably cuts viruses enough to meet EPA requirements to be rated as a purifier instead of a mere filter, but Sawyer isn't taking any chances. It is easy to back-flush to clean in the field. I love mine. Just scoop and drink.)
Boiling is easy, and kills everything, but takes time and fuel.
Ultraviolet water treatment (such as the SteriPen or Camelback units) is about to come of age. Current units are a bit heavy and theoretically fragile, but UV instantly kills virus, bacteria, and protozoans (as long as the water isn't too cloudy.) Downside- they need batteries. They'll be a much better option when UV-C LEDs become readily available and their weight and power requirements drop.
Hope this helped.