I wont get into the differences of how each filter filters out water mechanically, but they are different designs. Even though the absolute pore size differs, where the Sawyer mini is 0.1 microns and the Katadyn is 0.2 microns, there are other factors as well such as the silver in the Katadyn, etc. In the end they are quite on par with each other.

Advantage of the Katadyn is the build quality. You can likely throw this around and have it take a beating and it would still work (which is why they've been so popular for decades). The Sawyer has the rest really going for it - mainly the size and weight, which is really crucial to backpackers.

Why do I have both? I don't need both, really. I should drop the Katadyn to save weight. It is nice to have redundancy though, especially for such an important resource as water but I guess that depends how long you're planning to bug out for. I bought the Katadyn before the Sawyer Minis were a thing.

If I did it over again I'd likely combine the Sawyer Mini with a MSR MIOX Purifier Pen just for assurance. Filters are enough for North American waters, but you still have to watch out for contamination or chemicals. The pen would help a little bit more, since it also kills VX Nerve agent, among other chemical weapon formulations. Likely why it's used by the dept of defense. Doesn't hurt to have redundancy. Here is an old post of mine describing some of this

Btw, I'm not a big fan of the life-straws for this reason: shelf life of 3-5 years