Everyone has an opinion on which is better. Most people have a financial interest in one being better than the other. It really does depend on your situation but in my experience, the bags generally work better for a majority of people and here's why. You can buy replaceable bags and disposable bags. You won't need them as often as cartridge filter replacements. Bags also allow for better waterflow which means your cleaner is not working as hard, but it is filtering the same. This saves your cleaner from additional wear and tear, and allows it to use its power for cleaning, rather than forcing water through a dirty filter.
Both types will get clogged up depending on how much debris you throw at them, but most bag filters are 15-19 quarts, where cartridge filters are about 7-12 inches and 2 inches wide.
A bag filter can clean down to 2 microns, where most cartridge filters only boast a 50 micron cleaning level. 2 microns is about 1/20th the size of a human hair. This is a bacterial level clean, and is great for people with allergies or silt and algae in their pool.
The cartridge pool cleaner filter is easier to clean however, and makes emptying your cleaner a snap. If you have ever cleaned a filter bag you know what I mean. However, the invention of disposable filter bags makes disposing of filter contents easy. Just run the cleaner for a few cycles and throw it away. This is even easier than cleaning a cartridge filter. When they come out with a high waterflow disposable cartridge filter, I might change my mind but for now the bags work best.